GOP presidential candidates clashed over immigration, government spending and how to handle Russia's Vladimir Putin during Tuesday night's debate. As the night got heated, the audience got involved: here are the must-watch moments. (Ashleigh Joplin and Rebecca Schatz/Fox Business Network)

The fourth GOP presidential debate revealed deep splits among the eight candidates on the stage, especially on the subject of foreign policy — in which Sen. Rand Paul (Ky.) and billionaire Donald Trump made an unlikely tag team calling for skepticism about U.S. interventions overseas.

The two come from different places, philosophically: Paul out of a libertarian tradition personified by his father, longtime Rep. Ron Paul (R-Tex.), while Trump’s philosophy seems to be constructed as he goes along, revolving around the twin ideas that Trump makes better deals than anyone, and literally every problem is essentially a deal at its heart.

But the two of them both opposed other candidates — notably former tech executive Carly Fiorina, retired neurosurgeon Ben Carson and Sen. Marco Rubio (Fla.) — who called for more aggressive military policies.

“We can’t continue to be the policeman of the world,” Trump said.

“You can be strong without being involved in every civil war around the world,” Paul said.

From left,Marco Rubio, Donald Trump, Ben Carson, Ted Cruz, Carly Fiorina and Rand Paul take the stage. (Morry Gash/AP)

But their alliance was a short-lived one.

In Tuesday night’s debate, Trump faced harsh skepticism from others on the stage. Ohio Gov. John Kasich called Trump’s plans to deport illegal immigrants en masse “silly.” Fiorina jabbed him for saying he knew Putin because they had met in a TV green room. Even Paul, his ally on foreign policy, criticized Trump for not understanding the trans-Pacific trade deal, even as Trump attacked it.

The race’s other front-runner, Carson, had a relatively quiet debate again. But quiet debates have not hurt him so far.

“Thank you for not asking me what I said in the 10th grade. I appreciate that,” Carson said, after moderators made reference to questions raised about his telling of his life story. “I have no problem with being vetted. What I do have a problem with is being lied about. And putting that out there as truth.”

Both Paul and former Florida governor Jeb Bush had strong debates, interjecting themselves into ongoing arguments in a way they hadn’t before. But for both, there is no guarantee that a good debate will revive fortunes that have been sagging for months.

The Republican field split sharply on the question of how to handle the Islamic State and Russia, with Trump calling for the United States to stay out of more confrontations – and other candidates blasting him for advocating showing weakness.

“If Putin wants to go and knock the hell out of ISIS, I am all for it,” Trump said, noting that he had met Russian President Vladimir Putin through a “60 Minutes” episode. Trump expressed weariness about the troubles U.S. interventions had run into in Iraq and Libya. He also said the United States should not get involved militarily against the Islamic State, and instead let other countries in the region take the lead. “They say, ‘Keep going. Keep going, you dummies,’ ” Trump said. “We can’t continue to be the policeman of the world.”

Bush responded that Trump was misunderstanding how both Putin and the Islamic State see the world. “That’s like a board game. That’s like playing monopoly, that’s not how the world works,” he said.

That set off one of the most substantive exchanges of any Republican debate so far, which demonstrated a broad disagreement within the Republican field. Fiorina was among those calling for a more aggressive foreign policy, using both troops and diplomatic rebukes. She stepped in to criticize Trump for being naïve about the danger Putin posed, and said she herself would not talk to Putin right away. Although she said she had met him before: “Not in a green room for a show, but for a private meeting.”

Later, Sen. Ted Cruz (Tex.) and Kasich sparred about what they would have done if faced with the possibility that an enormous bank would fail. Cruz said he would not bail out the banks, as the federal government did during the financial crisis that began in 2008. Kasich said Cruz was showing he wasn’t ready to be an executive.

“When a bank is ready to go under, and depositors are ready to lose their life savings, you just don’t say, ‘We believe in philosophical concerns,’ ” Kasich said. He continued: “I’ve got to tell you, on-the-job training for president of the United States doesn’t work” – a jab that Cruz was as unprepared for the job as President Obama had been in 2009.

Cruz saw an opening. “So you would bail them out?” he said. That was a hard question for Kasich, since the bailout is deeply unpopular among many fiscal conservatives.

Kasich responded with a relatively vague promise to work out a plan, rather than simply repeating the same bailout that occurred back then. “You’ve got to deal with it. You can’t turn a blind eye to it,” Kasich said.

In general, Tuesday’s debate was one of the most lively and substantive of the primary season so far, with candidates delving deeply into their plans for taxes and spending. Moderators seemed in command of the facts behind their questions, and allowed arguments to run on – at one point, a moderator ignored the network’s own go-to-commercial music so that a back-and-forth could continue. The night’s main debate also featured standout performances by two candidates who sorely needed them: Bush and Paul. Some of the more prominent candidates in the polls took a back seat: Trump and Carson played minor roles, Rubio found himself on the defensive, and Cruz stayed out of the major back-and-forths.

Cruz’s most memorable moment might have been a mistake, in which he veered close to the error that undid fellow Texan Rick Perry in 2011. Cruz said he planned to cut five major agencies: the Departments of Commerce and Energy, the IRS, the Department of Commerce, and the Department of Housing and Urban Development. Commerce was in there twice, but nobody called him on it.

Earlier in the evening, Paul had attacked Rubio as a big spender who would put the country deeper into debt with new benefits and military spending, in one of the sharpest moments of the debate.

Paul, who had been largely invisible in the debates up to that point, suddenly took on Rubio, saying that when he added up Rubio’s tax plan and his military plans, “You get something that looks to me not very conservative.”

Rubio, whose best moment in these debates was a comeback, replied with an attack on Paul’s national security record.

“I know that Rand is a committed isolationist,” he said. “I’m not.”

Paul continued on the attack, casting himself in a role that he had seemed reluctant to play in the past. He was the one Republican who would argue that the U.S. could possibly spend too much on national defense.

“Can you be a conservative, and be liberal on military spending?” he asked.

Kasich, one of the lower-polling candidates on the main stage in the fourth Republican debate, also attacked Trump for advancing a “silly argument” about deporting 11 million illegal immigrants.

“Come on, folks, we all know you can’t pick them up and ship them back across the border. It’s a silly argument. It’s not an adult argument,” Kasich said, in one of the most direct challenges to Trump in this debate so far.

Trump responded by citing deportations – done on a smaller scale – under Dwight D. Eisenhower in the 1950s. “We have no choice,” Trump said. “We have no choice.”

When Kasich continued to press Trump, the billionaire responded with disdain: Trump said he runs a huge company. “I don’t have to hear from this man. Believe me,” he said. The crowd in the audience booed – showing a kind of disapproval that past debate crowds had usually reserved for the moderators.

Bush also rejected Trump’s call for deportations, saying it hurt the party’s ability to reach out to mainstream audiences: “They’re doing high-fives in the Clinton campaign right now when they hear this.”

Immigration was a dominant issue in the early going of the debate. Cruz jumped in to oppose Bush’s plan, calling it “amnesty.” He joked that party elites and the mainstream media were easy on Bush because they don’t feel the economic threat that immigrants pose to working-class Americans. If people were coming across the Rio Grande with journalism degrees, Cruz said, the American media would suddenly see immigration as a major problem.

 The debate began shortly after 9 p.m. with three candidates saying they oppose efforts to raise the minimum wage because they believe it would hurt low-wage workers more than it would help them.

“There is nothing that we do now to win. We don’t win anymore. … Taxes too high. Wages too high,” said Trump, adding that making the wage too high would hurt America’s ability to compete with overseas manufacturers. “I hate to say it, but we have to leave it the way it is.”

“Every time we raise the minimum wage, the number of jobless people increases,” Carson said, saying the effect was particularly noticeable among African Americans.

“If you raise the minimum wage, you’re going to make people more expensive than a machine,” Rubio said.

In the early minutes of the debate, one voice missing was that of Bush, the former front-runner. At one point, he cut off Kasich, who seemed to be interrupting a question meant for Bush.

“You’ve already made two comments, John,” Bush said. “I got about four minutes last debate, I’m going to get my question this time.”

Bush then spoke about his desire to undo federal regulations, and blasted President Obama’s handling of the economy. “Hillary Clinton has said that Barack Obama’s policies get an A. Really?” Bush said, taking an aggressive tone, and focusing – unlike last time – on the leading Democrat in the race, not other Republicans. “It may be the best that Hillary Clinton can do. But it’s not the best that America can do.”

Earlier in the evening, four lower-performing candidates met in the fourth “undercard” debate.

The earlier contest was dominated by a running argument between two low-performing candidates with vastly different plans to succeed. One was New Jersey Gov. Chris Christie, who is aiming to attract moderate voters, and the other was Louisiana Gov. Bobby Jindal – who is trying to convince conservatives that moderates like Christie are untrustworthy sellouts.

Jindal, now consigned to the low-performers’ debate for the fourth time, was on the attack from the beginning of the debate. Christie, just relegated from the main stage, was an uncooperative target. Jindal repeatedly zinged Christie for being a “big-government Republican,” and at one point said he should get a “juice box” for participation in the conservative movement.

Christie, showing an uncharacteristic discipline in the face of annoyance, repeatedly said he was more interested in beating Hillary Clinton than in battling with Jindal. Christie – a former federal prosecutor – used the verb “prosecute” repeatedly when describing how he could debate Clinton in the general election. “You need someone who’s going to stand up on that stage and prosecute the case against her,” he said. Christie also made an appeal to law enforcement officers, saying he would support them more than President Obama had in the roiling debate over police shootings and police brutality.

“I will have your back,” Christie said.

Christie also ended on a note of bipartisan hopes: “I will go to Washington. ... To bring this entire country together for a better future for our children and grandchildren.”

The other two candidates on stage had less air time but still managed memorable moments. Former senator Rick Santorum (Pa.), describing his grudging admiration for hardball tactics employed by Democrats, shouted out the words “They fight!” It was the loudest moment of the night, though an odd message to leave people with.

Former Arkansas governor Mike Huckabee, who’s also been relegated from the main stage, once again defended federal “entitlement” programs like Medicare and Social Security, saying that Americans had paid into those programs and deserved to have them pay out. His most memorable moment, however, was a joke. When asked about Federal Reserve Chair Janet Yellen, Huckabee responded like the folksy preacher he used to be: “Well, my wife’s name’s Janet. When you say Janet yellin’, I’m very familiar with what you mean.”

At various points, Jindal called out Cruz – though not by name – as all talk and no action. He called out Huckabee for being a big spender. And he zinged Christie, who had been the dominant figure in the debate’s first half, for doing too little to cut state spending.

“Let’s not be a second liberal party,” Jindal said, talking about Christie. “Let’s just not beat Hillary. Let’s elect a conservative to the White House, not just any Republican.”

Christie, for his part, did not rise to Jindal’s bait. Instead, he repeatedly cast himself as somebody who could appeal to moderates and win in blue states.

The most memorable moment for Santorum came when he tried to play off Jindal and Christie’s tension. He noted that one man claimed to be a true conservative, and the other said he could win in a blue state. Santorum raised his arms in a “Why not me?” gesture, a funny moment that indicated that he thought he combined both qualities.

Christie and Huckabee also spent time attacking the IRS and blaming the federal tax code for weakening manufacturing and America generally.

“First, make the tax code fairer, flatter and simpler,” said Christie. He described making the U.S. tax code so simple that individuals could file their taxes in 15 minutes. “I’ll be able to fire a whole bunch of IRS agents once we do that,” Christie said.

Huckabee went Christie one better and said that he’d eliminate the income tax entirely, and move to a “Fair tax,” essentially a national retail sales tax.

“We get rid of the IRS. We completely eliminate it. Because the government has no business knowing how much money we make, and how we make it ... that’s none of their business,” Huckabee said.

Huckabee and Christie were dismissed from the main debate stage after not reaching the 2.5 percent threshold in national polls. They joined Jindal and Santorum, who have been in the low-performers’ debate all along.

Two other candidates — former New York governor George Pataki and Sen. Lindsey O. Graham (S.C.) — were bumped from the undercard entirely after failing to reach a lower threshold of 1 percent in national polls. (Former Virginia governor Jim Gilmore, who appeared in the first undercard but not the next two, will again be kept off the stage because of low numbers). Graham, with a folksy charm and an aggressive call for greater military intervention, had been the dominant candidate in both of the last two undercards. It was not enough to raise his poll standing.

These debates come just 13 days after the last Republican debates, which were widely perceived as a disaster for their hosts at CNBC. Both the crowd and the candidates attacked the moderators for asking argumentative questions, and the moderators let the candidates get away with blatant dodging of the truth. Trump, for instance, claimed he hadn’t said something that was included in his own immigration policy paper. The moderator, who had been right, apologized.

Tuesday night’s main event was moderated by Fox Business personalities Maria Bartiromo and Neil Cavuto. In an interview with The Washington Post, they said they aimed to do better — and that they wouldn’t put up with whining.

“I understand candidates getting annoyed, but they better be careful about looking like whiners and babies. I see this on the right and the left. I think you can ask very tough questions without coming off like an ass,” Cavuto said in an interview with Callum Borchers of The Fix. “I think it’s incumbent on us to know and appreciate the difference.”

Ahead of the debate, Republican National Committee Chairman Reince Priebus sent a letter to RNC members saying the committee has “worked tirelessly” with Fox to “fine-tune our debate process and ensure that our candidates have the best format and experience.”

“I am confident that we will finally have a debate focused on the economy and financial matters. The moderators tonight have also pledged to allow candidates 90 seconds to answer questions, rather than 60 seconds in past debates, as well as give each candidate a closing statement,” Priebus says in the letter, which was shared with The Washington Post by a Republican recipient.

Democratic front-runner Hillary Rodham Clinton issued a prebuttal ahead of the debate in the form of an Internet video that accused Bush, Rubio and Trump of backing tax plans that disproportionately benefit the wealthy.

The last debate was a breakout moment for Rubio. He was attacked by Bush — an old ally of Rubio’s from their days in Florida politics — for missing many votes in the Senate.

Rubio responded with a deeply stinging cut-down: “The only reason why you’re doing [this] now is because we’re running for the same position, and someone has convinced you that attacking me is going to help you.”

Sean Sullivan, Jenna Johnson in Springfield, Ill., Philip Rucker in St. Francis, Wis., and Robert Costa in Milwaukee contributed to this report.