Thomas E. Mann is a senior fellow at the Brookings Institution. Norman J. Ornstein is a resident scholar at the American Enterprise Institute. They are the co-authors of “It’s Even Worse Than It Looks: How the American Constitutional System Collided With the New Politics of Extremism.”

So what happens when a party that has defined itself as an insurgent outlier, scornful of compromise and dismissive of the legitimacy of its opposition, actually takes charge in Washington?

The Republicans now control both the Senate and the House. The people have spoken — and loudly. Mitch McConnell is exultant, finally achieving a lifetime goal. The new majority leader will get a wonderful office with a better view of the Mall, a nicer car and the ability to control the agenda on the Senate floor. But he will also get a deeply divided party and a constant headache trying to keep all of his colleagues in town to vote, especially with several of them on the presidential campaign trail, raising cash and appealing to the most extreme elements of the base.

John Boehner, enjoying perhaps the largest Republican majority in the House since 1929 , will also face serious challenges. The incoming lawmakers are more radical and anti-establishment, and Boehner will have to work harder than ever to keep even this more robust majority in line for anything save hard-line legislation.

Welcome to the 114th Congress, in which the warfare within the GOP will only be amplified by the party’s new power. The pragmatic desire of mainstream Republicans to transcend their “party of no” label and show that they can actually govern will clash with the forces that continue to pull the GOP to the right and oppose anything the president does. This fight within the party will define the new Congress nearly as much as the battles with a Democratic president.

During the 2014 cycle, Republicans ran disciplined campaigns for the Senate — there were no candidates along the lines of Todd Akin or Sharron Angle to sabotage the party’s larger ambitions — but a disciplined anti-Obama message hardly means that the GOP establishment has beaten back the insurgency. McConnell and his fellow candidates talked after their victory about coming together to govern, but they’re also on record pushing to repeal Obamacare and roll back other core Obama policies. If anything, the breadth and depth of the Republican victory will convince the party base — and the conservative activists, talk-radio hosts and bloggers animating it — that the obstruction of the past several years worked beautifully, that they have the power and the mandate to push radical anti-government policies, and that any compromise would be abandonment and betrayal.

In the Senate, even candidates such as Iowa’s Joni Ernst and Arkansas’ Tom Cotton, who had the blessing of the GOP mainstream, embraced nearly every policy idea and conspiracy theory of the tea party wing. And you only need look at Sen. Marco Rubio’s repudiation of his own immigration reform bill to see where the party is headed.

McConnell’s hope for a return to the Senate of old will prove an illusion. Just wait until the first time he brings up a bill — say, to approve the Keystone XL pipeline — and Democrats offer a string of amendments designed to embarrass the 24 Republican senators up for reelection in 2016. Before very long, McConnell will be “filling the amendment tree” to protect his colleagues, just as Sen. Harry Reid, his nemesis across the aisle, did. And as Democrats use the filibuster to block the GOP strategy of forcing the president to sign or veto a barrage of bills, McConnell will face pressure to employ his own “nuclear option” and eliminate the filibuster for bills.

In the House, Republicans have added to their majority — but it is a chamber more starkly polarized than before, with fewer loyalists to the Republican leadership and fewer Blue Dog Democrats to step in if there is a vacuum. Boehner is losing allies and party stalwarts, such as Tom Petri of Wisconsin, to retirement. December brings leadership contests; Boehner will survive, but any mainstream desire to come up with a positive health-care reform alternative, immigration legislation or even an infrastructure bill will run into a tea party buzz saw. (Immediately after the election, Boehner moved from conciliation to predicting a House vote against key parts of Obamacare, only emphasizing that tension.) And finding compromises with the Senate under those circumstances — on bills that can either win 51 Republican votes or survive Democratic filibusters — will be an uphill fight.

All this complicates the GOP establishment strategy heading toward 2016, as articulated by House Majority Leader Kevin McCarthy, McConnell, Republican Party Chairman Reince Priebus and others: Push the president to cede ground on things such as Obamacare’s medical-device tax and the Keystone pipeline; meanwhile, pass a set of bills that show a constructive Republican agenda, make the president veto them and then identify Democrats as the party of obstruction.

Moreover, the cultural and structural differences between the House and the Senate ensure nearly as much conflict between them as between Congress and the president. When Republicans swept into majorities in both houses in 1994, the two bodies were totally out of sync. Newt Gingrich’s House quickly passed its “Contract With America” and waited for Bob Dole’s Senate to follow suit, so that together they could force President Bill Clinton to sign or veto the popular, poll-tested ideas. But the Senate Republicans had a different time frame and different ideas, not to mention concerns about Democratic filibusters. Before long, Gingrich was complaining about Dole — and Dole responded in kind.

Then and now, a Republican House and a Republican Senate do not naturally translate into a spate of Republican legislation. There is a reason why Republicans have not come up with a clear alternative to Obamacare, why they discuss entitlement reform only in general terms and why they squashed the comprehensive tax reform proposal from Rep. Dave Camp, the House Ways and Means Committee chairman, before the election. It is just not easy to come up with concrete policies aimed at solving problems without creating schisms in the party’s ranks or an outcry from its base.

There are things the new Republican Congress will be able to do: block most of Obama’s judicial nominations and many key executive ones, hold countless hearings to investigate scandals real and imagined, harass Obama officials by demanding that they testify repeatedly, and issuing subpoenas for documents and records that will tie up the White House counsel’s office. Those actions will make it more challenging for Obama to use his executive authority as fully and expeditiously as he would like.

Of course, the conservative base wants more than that. Activists want to reverse or eviscerate Obama’s accomplishments, from health-care reform to financial regulation — in other words, not just flesh wounds but fatal blows to the president’s legacy. The only real way to do that is through the power of the purse. Indeed, McConnell has promised to use budget reconciliation (the procedure that can be passed with simple majorities and no filibuster) to weaken Obamacare and undercut the Dodd-Frank financial reforms, among other things. But if a bill that did more than make token adjustments to his top achievements went to the president, he would veto it, setting up another deep confrontation that could lead to a government shutdown in October. McConnell went out of his way Wednesday to insist that there would be no government shutdown, but that pledge conflicts with the goal of using the budget to bring Obama to his knees.

How does a two-term president entering his final stretch in office deal with this polarized, even tribalized political environment? Obama faces not only a fully Republican Congress, but a restive Democratic base already angry about issues from drone strikes to the failure to move on immigration reform and now fearful that their leader will cut deals with the enemy.

The opportunities to get some things done with GOP participation will be few and far between. Obama will need to balance his professed willingness to work with the Republicans with a realistic — and aggressive — use of his executive power. In his post-election news conference Wednesday, the president expressed openness to GOP ideas but did not shrink from the use of executive orders.

One way to do so is on immigration reform. The president had promised to act right after the election, a pledge he reiterated this past week. But immediate unilateral action following a big GOP victory would prompt a Republican outcry — and promises to obstruct Obama even further. A possible solution: Issue an executive order that curtails deportations and expands the number of “dreamers,” but delay its implementation until the end of January, with a challenge to congressional Republicans to send him a comprehensive immigration reform bill before then to head it off. On climate change, the president might couple EPA regulations to limit carbon emissions from existing power plants with presidential approval of the Keystone pipeline, thus mollifying his base while co-opting a Republican initiative. And on trade deals, sentencing reform, corporate tax reform, patent reform and overhaul of the National Security Agency, Obama can reach out to Republicans early on to show his bipartisan bona fides.

More broadly, Obama must embrace his role as explainer in chief, defining what his presidency and his party stand for and how that contrasts with Republicans, describing the destructiveness of divided government under present conditions, and stressing a plausible path to sustained and widely shared economic growth. And do all that, of course, while navigating a bitter battlefield that may even include an effort from the conservative base to impeach him.

GOP figures in the Senate such as Tennessee’s Bob Corker and Arizona’s Jeff Flake, bolstered by conservative intellectuals, are trying hard to create a new center-right space in the party. But they’re far from overcoming the larger forces at work. Until one party controls both ends of Pennsylvania Avenue for more than two years, or the Republican Party as a whole feels more powerful incentives to reengage in negotiation and compromise, there is no reason to expect warfare to wane or governance to prevail.

Read more from Outlook and follow our updates on Facebook and Twitter.