WASHINGTON — Much of White House hopeful Mitt Romney’s fortune lies well-hidden in a network of opaque offshore investments including some $30 million in the Cayman Islands, Vanity Fair reported Tuesday.

Romney has amassed vast wealth — estimated to be as high as $250 million — since founding private equity firm Bain Capital in 1984, and he consistently says his successful business experience is what puts him in better position than President Barack Obama for turning around the sluggish US economy.

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But while his campaign insists Romney has not exploited the offshore havens to avoid paying necessary US taxes, the difficulty in tracking the overseas transactions and holdings raises questions about the candidate’s finances in the heat of a campaign.

The report detailed how Romney continues to have personal interests in at least 12 of the 138 funds organized by Bain in the Caymans, where such investments are hidden behind confidentiality disclaimers, making an assessment of Romney’s true wealth virtually impossible.

He also holds a Swiss bank account — with $3 million in it, according to 2010 tax returns — and other interests in tax havens such as Bermuda, according to the report in the August issue of the magazine.

Romney’s tax rate has been a particular point of contention. In 2010 he reported income of $21.7 million, mostly from investments, and paid just over $3 million in taxes, a paltry rate of just 13.9 percent, far lower than the rate for most middle-income Americans.

Most of Romney’s money is earned in investments, which are taxed at just 15 percent, compared to the 35 percent he would pay on wages.

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Many of those investments are offshore, with 55 pages of Romney’s 2010 tax return devoted to his transactions with foreign entities, according to Vanity Fair.

“What Romney does not get,” veteran Washington lawyer and offshore expert Jack Blum told the magazine, “is that this stuff is weird.”

The White House jumped on the report, describing it as further evidence of Romney’s “bets against America” and suggesting the story about mystery corporations and offshore counts leads to serious questions.

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“The question is, why? Was he avoiding paying his fair share of US taxes?” Obama campaign spokesman Ben LaBolt wrote in an email.

“Was he hedging against the dollar? Until he releases his tax returns from that period, Americans will never know.”

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Romney has refused to release pre-2010 tax returns.

“Mitt Romney’s economic philosophy has always put maximizing his profits above anything else,” LaBolt added.

The Romney campaign released a statement on the article, but it was a reaction to the Obama campaign’s treatment of the story and did not address its specifics.

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“As job growth slows, manufacturing activity stalls, and our economy continues to sputter, President Obama knows he can’t make a legitimate argument for another term in office, so instead he is trying to tear down his opponent,” Romney spokeswoman Andrea Saul said.