Ben Carson is a renowned pediatric neurosurgeon and Republican contender for the White House in 2016. Here's his take on Obamacare, homosexuality and more, in his own words. (Julie Percha/The Washington Post)

After last month's primary debate on Fox News, Republican voters started paying attention to Ben Carson. The retired neurosurgeon's ratings ballooned, and in a poll published Monday by The Washington Post-ABC News, 20 percent of Republican-leaning voters said they'd support him if the primary were held today.

This week, as the Republican candidates return to the stage for their second debate, Carson is the runner-up behind Donald Trump, with 33 percent.

Between Carson's and Trump's supporters, a majority of Republican voters at this moment stands behind two unconventional candidates. Neither man has any political experience. Nor do either of them have the kind of polish or charisma common to other politicians. Trump is brash, while Carson is soft-spoken. To the extent that disillusionment with professional politics is the source of the pair's support, it is easy to imagine Carson's supporters voting for Trump instead. If Carson dropped out of the race and they did, the real-estate magnate could win the nomination.

[Read more: Trump, Carson top GOP race; Clinton leads Dems but support drops]

The most surprising thing about this duo, however, might be the nuance in their agendas.

Trump has suggested raising taxes on rich Americans. He and Carson both have complicated views on abortion. Neither has won over the Republican base by being the most conservative candidate in the field. Maybe conservative voters are more willing to forgive deviations from the party's orthodoxy than the Beltway establishment has recognized.

You can read more about Trump's platform here and here. The post below explores Carson's positions on abortion, taxation, and more.

Taxes and the budget

One important issue on which Carson is arguably the most conservative candidate is the federal budget. Carson supports a version of the flat tax advocated by some of his rivals. Under his system, all Americans would pay 10 percent of their income in taxes ("although it could be any percentage," he has written). There would be no exemptions for the poor. Former Arkansas Gov. Mike Huckabee advocates a similar system, but he has promised that the poor would be spared paying taxes. Even libertarian Sen. Rand Paul (R-Ky.) has called for a flat tax on all income above $15,000, giving low-wage and part-time workers a break.

Like Paul, Carson has also advocated amending the Constitution to require Congress to balance the federal budget every year, which would mean that the government would be unable to borrow money and could only spend what it collects in taxes.

[Read more: Why Rand Paul wants Congress to balance the budget every year]

Carson's tax plan, which is based on the biblical tithe, would raise roughly $1.1 trillion in revenue, according to an analysis by Citizens for Tax Justice. That sounds like a lot, but it really isn't much. The federal government will raise $3.5 trillion and spend about $3.9 trillion next year, according to the Congressional Budget Office.

Carson's positions on the flat tax and the balanced budget amendment together imply an immediate 72 percent reduction in federal spending, basically the equivalent of eliminating Social Security, Medicare, Medicaid, food stamps and the entire military.

Abortion

On abortion, Carson's views are less clear. He has conducted medical research using tissue from aborted fetuses and referred female patients to abortion clinics, Politico's Katie Glueck reports.

"He believes in people making their own decisions based on facts and information," Carson's spokesman told her. "He has always believed that the battle over abortion had to be waged in the hearts and minds of Americans, that you cannot legislate morality."

That statement suggests that Carson believes abortion should remain legal, even though he is personally opposed to it. In other words, Carson's views on abortion are similar to Vice President Biden's. "I've been a practicing Catholic my whole life," Biden said in 2012.

"Life begins at conception," the vice president continued. "That's the church's judgment. I accept it in my personal life. But I refuse to impose it on equally devout Christians and Muslims and Jews and — I just refuse to impose that on others."

Carson apparently agrees with Trump as well. Trump has also said that while he is opposed to abortion personally, he is "very pro-choice."

Banking, guns and more

On other issues, Carson has surprisingly liberal views, as Timothy Murphy reports in Mother Jones.

Despite writing in praise of the Second Amendment, Carson has also suggested that he consider a ban on semi-automatic and assault weapons in urban areas.

"I've lived in urban areas. I've worked in urban areas. I've seen a lot of carnage, and I'd prefer a situation where the kinds of weapons that create that kind of carnage don't fall to the hands of criminal elements or insane people. But that is secondary to the desire to always defend the Second Amendment," he later clarified.

And in a book published last year, Carson specifically mentioned the Glass-Steagall Act, a law passed after the Great Depression separating old-fashioned, Main-Street banking from the riskier and more profitable business done on Wall Street. Progressive policymakers such as Sen. Elizabeth Warren (D-Mass.) have argued that when he repealed Glass-Steagall, President Bill Clinton helped create the conditions for the most recent financial crisis.

[Read more: Elizabeth Warren and John McCain want Glass-Steagall back. Should you?]

"The stock market crash of 1929 exacted a severe toll on the people of our nation and our legislators realized, in hindsight, that some of our banking and investment policies had contributed to the crash," Carson wrote in One Nation. "Seventy years later we forgot about many of the horrors of those difficult financial times as well as the reasons why we imposed appropriate regulation on speculative financial activity."