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The Romney camp later walked back his remarks, telling The Washington Times that the only path he has identified would be for illegal immigrants to join the U.S. military.

Whatever the case, Mr. Romney began softening his stance on immigration — a perennial top issue for Hispanic voters in polls — before the first debate, telling The Denver Post something that he refused to say for months: He would not immediately deport the illegal immigrants granted temporary legal status under Mr. Obama’s nondeportation directive.

He also told the The Des Moines Register’s editorial board a few days later that “there is no legislation with regards to abortion that I’m familiar with that would become part of my agenda” — a direct appeal to female voters in swing states, who in a recent Gallup Poll said abortion is the most important issue for them this election season.

Along the way, Mr. Romney also has generated some doubt about whether he supports the National Defense Authorization Act after saying in a debate earlier this year that he would in fact sign the legislation as written.

Ambiguity as a strategy

Mr. Romney’s vagueness on some of these issues has irritated some people.

Michael F. Cannon, director of health policy studies at the libertarian-leaning Cato Institute, said he had decided “there’s no point in trying to nail [Mr. Romney] down” on what he will do with Obamacare.

“He won’t be nailed down; he wants ambiguity. And even if you did, his word is a poor predictor of his thoughts and future actions because he’s Mitt Romney,” Mr. Cannon said.

Frank Sharry, executive director of America’s Voice, which advocates for immigrant rights, said that when it comes to immigration, Mr. Romney is “trying to put lipstick on the pig.”

“The pig, his policies, amount to this: All but young undocumented immigrants who serve in the military should be forced out of the country. His rhetoric is this: I want permanent solutions on a bipartisan basis,” Mr. Sharry said. “He’s got the fog machine on in hopes that Latino immigrants will think he’s actually going to do something for them. But they get this issue better than the Boston brain trust. They get he actually wants to do something to them.”