President Barack Obama's re-election campaign on Tuesday redoubled its attack on Mitt Romney as someone who shipped jobs overseas, defying findings from independent fact-checking outlets that have branded the accusation largely baseless.

In a new television ad, dubbed "Believe," Team Obama says that companies taken over by Romney's Bain Capital "were pioneers in outsourcing U.S. jobs to low wage-countries" and that the Republican standard-bearer himself "supports tax breaks for companies that ship jobs overseas."

The commercial highlights Obama's rescue of the auto industry—which Romney opposed—and says the president would strive to bring companies back to American soil. "Outsourcing versus insourcing. It matters," the ad's narrator says.

The message will run in Colorado, Florida, Iowa, North Carolina, New Hampshire, Nevada, Ohio, Pennsylvania and Virginia, according to the campaign. And Obama is sure to hit that note as he campaigns by bus through Ohio and Pennsylvania on Thursday and Friday.

The Romney campaign struck back quickly at what it called "untrue ads." "We are happy to put Gov. Romney's record of job creation in the private sector, and as governor, up against President Obama's any day," said spokeswoman Amanda Henneberg.

The new commercial came a day after the Obama campaign picked a very public fight with the independent group FactCheck.org—and drew a sharp rebuke for its trouble.Deputy campaign manager Stephanie Cutter wrote to argue that the campaign's previous outsourcing charges against Romney and Bain were correct. Factcheck pored over Cutter's six-page complaint and concluded that "it cobbles together selective news snippets and irrelevant securities documents" to make its argument. "In a nutshell, the Obama campaign is all wet on this point."

But recent polls in pivotal battleground states have found that the incumbent's approach may be working. And it's unclear whether the average voter will distinguish between Bain Capital under Romney's management and what happened after he left the company in 1999 to run the Salt Lake City Olympic Games—especially since he continued to benefit financially from Bain's business decisions. There's no sign that the Obama campaign will change its game plan, particularly since his team aims to make the election a choice between the candidates' economic visions—while Romney's campaign hopes voters will view November as a referendum on whether they are better off now than they were when the president took office.