(RNS) Americans want their presidents to be religious, but many have trouble identifying the faiths of President Obama and leading GOP contenders Mitt Romney and Rep. Michele Bachmann, according to a new poll released Monday (July 25).

A majority of Americans (56 percent) say it's important for a candidate to have strong beliefs, even if those beliefs differ from their own, according to the poll conducted by Public Religion Research Institute in partnership with Religion News Service.

Yet the religious groups most firmly behind this point -- white evangelicals (73 percent) and ethnic minority Christians (74 percent) -- often falter when asked about politicians' religions.

For instance, just 44 percent of white evangelicals know that Romney is a Mormon. At the same time, more than 8 in 10 evangelicals say Mormon religious beliefs greatly differ from their own.

Even fewer ethnic minority Christians (21 percent) knew Romney's religion. And only one in three Americans can correctly identify Obama's Christian faith. Consistent with previous polls, about one in five (18 percent) Americans think Obama is Muslim.

Daniel Cox, the research director at PRRI, said Romney's Mormonism could be a liability: of people who say Mormon beliefs are significantly different than their own, Obama currently leads Romney, 49 percent to 28 percent.

"Because views about the Mormon faith are tied to political support, Romney will need to address these perceptions as Americans learn more about him during the campaign," Cox said.

Still, observers cautioned that perceptions can change over time, or even take a back seat to other factors like party loyalty or pocketbook concerns. And for Republicans, simply beating Obama could be the most important factor of all.

White evangelicals "are going to be more likely to vote Republican, even if the party nominates someone who isn't known for strong faith commitments," said Gary Scott Smith, an expert on presidential religions at Grove City College in Pennsylvania.

"And if they don't recognize that Romney's a Mormon by now, then you wonder how attuned they are to politics anyway."

In other findings:

-- Just four in 10 Americans can correctly identify Romney's religion; 46 percent say they don't know. When asked Obama's religious beliefs, a full 40 percent of Americans say they didn't know.

-- White evangelicals are the group most likely to say they don't know what Bachmann's beliefs are (51 percent), even though she attends a Baptist church, and only 35 percent say she has similar religious beliefs to them.

-- At a little more than 70 percent, Republicans and Tea Party members are significantly more likely than Democrats (51 percent) to say it's important for a presidential candidate to have strong religious beliefs. Tea Party members (46 percent) are even more likely than Republicans as a whole (38 percent) to say it is "very" important for a candidate to have strong religious beliefs.

-- People who say it is important for a candidate to have strong religious beliefs tend to prefer Romney to Obama, 43 percent to 36 percent, or Bachmann over Obama, 44 percent to 38 percent, in head-to-head matchups.

Americans have traditionally elected presidents who use religious language and seek divine guidance, especially when grappling with the moral conflicts of the day, provided that their beliefs are relatively mainstream and don't conflict with national security, Smith said.

The appeal of a visibly devout candidate, even if those beliefs aren't actually understood, also reflects some mistrust of our political system, said Mark Silk, a professor of religion in public life at Trinity College in Hartford, Conn.

"There's a suspicion of a strong central government -- you can see that in the Tea Party -- and Americans think that more religious leaders are less likely to abuse the people," he said.

Obama has walked a careful line on his religious beliefs -- talking openly about "glory(ing) in the promise of redemption in the Resurrection," for example, but also trying not to alienate secular voters. If, after four years, six in 10 Americans still don't know he's a Christian, there may be little he can do to change the perception.

Bachmann, however, may have an opportunity to gain votes among fellow evangelicals by continuing to publicize her religious convictions, while Romney may be better off keeping undecided voters focused away from his Mormon faith, Silk said.

The PRRI/RNS Religion News Survey was based on telephone interviews of 1,012 U.S. adults between July 14 and 17. The poll has a margin of error of plus or minus 3 percentage points.