A United Auto Workers health-care trust that holds  a 41.5% stake in Chrysler -- a legacy of the automaker's 2009 bailout -- said on Tuesday that it wants Fiat (NASDAQOTH:FIATY) to pay almost $343 million for a piece of its stake.

That's more than double Fiat's $139.7 million offer for the piece, which represents 3.3% of Chrysler.

The UAW trust says Fiat's offer is "substantially below market value." But Fiat, which is tying its hopes to an interpretation of legal documents drawn up in the wake of Chrysler's bankruptcy that give Fiat options to buy Chrysler over time, may not have the cash to up its offer.

What's going on? And how is this likely to play out?

A legacy of Chrysler's bailout
The story here starts with the Obama administration's packaged bailout of Chrysler and General Motors (NYSE:GM). With GM, as most investors know, the U.S. Treasury gave a loan of almost $50 million in exchange for, in part, a big block of GM stock -- some of which the government still holds.

Chrysler's deal was different. The smallest of Detroit's automakers was in dire shape after years of mismanagement by a succession of increasingly inept owners, and more drastic action was needed. After much wrangling, ownership of Chrysler was split between the UAW's trust and Fiat, with the Treasury retaining some shares.

Fiat was given operational control -- Fiat CEO Sergio Marchionne became Chrysler's CEO as well -- and, importantly, the option to acquire more of Chrysler over time. When Fiat paid off the last of Chrysler's bailout loans and bought the Treasury's shares last year, it officially became Chrysler's majority owner.

But it doesn't yet own enough to merge Chrysler with Fiat -- which would allow cash-strapped Fiat to access Chrysler's cash hoard.

That's where the UAW comes in.

Funding retiree health care with Chrysler stock
The UAW health-care trust, called a "VEBA," or "voluntary employee benefit association," is a legacy of old Detroit. In the old days, the Big Three's agreements with the UAW included private health insurance for automakers and their families after they retired.

That was an expensive benefit, and the Detroit automakers desperately needed it to go away to rein in their costs and become globally competitive. The deal that the automakers struck with the union was this: The benefit would go away for future retirees, but the automakers would contribute to trust funds that would continue to pay for insurance for existing retirees.

Ford (NYSE:F) funded  its VEBA on its own, at the end of 2009. Chrysler's, on the other hand, was effectively funded as part of its bailout -- with Chrysler stock. It was always the plan that Fiat would buy the stock eventually, provided that it was able to successfully manage Chrysler's business.

By any measure, Fiat has been very successful with Chrysler. Chrysler's U.S. sales have risen dramatically -- up  22.5% this year through October, on top of huge gains in 2011 -- on the strength of a radically improved product line, the result of a skilful high-speed overhaul instituted by Marchionne shortly after Fiat assumed control. Chrysler has generated strong profits in recent quarters -- in contrast to Fiat's ongoing struggles in Europe.

But the question now is this: In light of Chrysler's success since 2009, what's a fair price for the stock held by the VEBA?

Figuring out a fair price for the UAW's stake -- one way or another
That question is now going to court. The calculation used to determine the value of the VEBA's Chrysler shares is set forth in documents drawn up during the bailout -- but Fiat filed a lawsuit arguing that a clerical error in those documents messed up the formula. The difference is nontrivial: It's the difference between Fiat's $139.7 million offer for 3.3% of Chrysler, and the $343 million that the UAW is demanding instead.

Compounding the problem is that Fiat is short of cash. While Marchionne would dearly love to acquire 100% of Chrysler, and had cash set aside to do so, Fiat has been forced  to spend big to shore up its operations in recession-ravaged Europe.

At the same time, the folks running the VEBA are eager to sell, to diversify their holdings -- but they want to get a good price. If Fiat won't up its offer, the VEBA can (and probably will) take Chrysler public with an IPO. Stay tuned.

Fool contributor John Rosevear owns shares of General Motors and Ford. Follow him on Twitter at @jrosevear. The Motley Fool owns shares of Ford. Motley Fool newsletter services have recommended buying shares of General Motors and Ford. Try any of our Foolish newsletter services free for 30 days. We Fools don't all hold the same opinions, but we all believe that considering a diverse range of insights makes us better investors. The Motley Fool has a disclosure policy.