TAMPA, Fla. — The Republican Party that’s showing its face to America this week is a restless institution that relies heavily on the uncompromising passions of tea partyers, anti-immigration activists and social conservatives. It’s a potent but unruly coalition that supplies vital energy today but poses serious challenges for the future.

These forces propelled the GOP to big wins in 2010, and they might help Mitt Romney win the White House this fall. But they operate largely beyond his control, sometimes igniting brushfires and pulling his campaign off message.

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Reince Priebus, chairman of the Republican National Committee, speaks to delegates during an abbreviated session of the Republican National Convention in Tampa, Fla., on Monday. The Associated Press

More troubling for the Republican Party in future elections is that these fiery conservatives seem to be turning off many Hispanic voters, the fastest-growing segment of the American electorate.

The challenge facing GOP leaders as they hold their nominating convention and look to the future: trying to win elections and push their agenda through Congress by harnessing the energy of these conservatives without letting that energy turn on them — and without letting it put the party badly out of step with a nation that’s rapidly becoming less white.

In Romney, the convention-goers in Tampa are nominating a former corporate executive who fits somewhat uneasily in the party’s decades-long rightward shift, which manifests itself most clearly in Congress.

Social conservatives remain wary of Romney, who once backed abortion rights and gun control. Their antipathy toward President Obama, however, may well nudge them along to polling places in November.

That is Romney’s great hope, even as he and his GOP establishment allies try to dampen some of the political right’s most vocal figures. Romney and others urged a Missouri congressman to drop out of the Senate race after he made widely condemned remarks about rape and pregnancy.

Romney’s words angered some religious activists who see the congressman, Todd Akin, as a champion of anti-abortion efforts.

On the economic front, Romney is in step with his party’s anti-tax advocates. He promises to cut taxes further, saying that will help strengthen the economy despite some economists’ warnings that tax cuts have contributed heavily to the nation’s massive debt.

GOP lawmakers’ adamant opposition to tax hikes, even on the wealthiest families, puts them at odds with most Americans.

But much more problematic is the Republican Party’s strained relationship with minorities, especially a fast-growing Hispanic population alarmed by the sometimes sharp tone of conservatives on illegal immigrants. The party may need to address that problem before long to avoid falling behind Democrats in key states.

For now, Republicans have a right to party with some swagger in Tampa. Strategists in both parties say Romney has a solid chance of defeating Obama.

In congressional races, Republicans are well-positioned to keep their House majority. And they might pick up the handful of Senate seats they need to take control of that chamber, too.

Should those three things happen, Republicans will be able to reshape much of the government’s tax, spending and regulatory policies to their liking, checked only by Senate Democrats’ ability to sustain filibusters here and there. Republicans presumably would dismantle much of “Obamacare,” extend all the Bush-era income tax cuts, reduce other levies including the corporate tax, cut federal payments to Medicaid and loosen regulations on banks and other businesses.

But whether Romney wins or not, his party’s congressional leaders will continue to struggle to keep a lid on their tea party wing. Several more Republicans who vow not to compromise with Democrats are headed to Congress, and they will give new headaches to House and Senate leaders seeking ways to pass legislation in a nation that’s split almost 50-50 politically.

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