The Obama administration will take its first stab at this vexing question Tuesday. Treasury’s Conference on the Future of Housing Finance will bring together bankers, investors, housing experts and policymakers in search of a fix for the government-sponsored mortgage investors, which have consumed nearly $150 billion of taxpayer support since their September 2008 takeover.

Fannie and Freddie don’t make loans, but they guarantee mortgage loans made by others and purchase some loans for their own portfolios. They extend credit to homebuyers by making financing available to lenders, and keep mortgage markets liquid by making mortgage purchases.

By extending credit with an implicit government guarantee, the companies create huge risks to taxpayers, as we have seen. But to make changes now, at a time when the economy appears on the verge of another damaging downturn, is to risk pulling the rug out from under an already unsteady recovery.

So what to do?

Fannie and Freddie “are quite profoundly broken,” says Raj Date of the Cambridge Winter Center. “But no one wants to disrupt the only thing that’s working right now in the mortgage market.”

And as usual, there is little agreement over what shape a cure should take. Treasury Secretary Tim Geithner has said he believes there is a “strong economic and public policy case” for retaining a government guarantee of mortgage lending under the right circumstances. Pimco’s Bill Gross says he wouldn’t buy the companies’ debt without one.

But the Republicans in the House insist the companies simply should be privatized, however unrealistic that is. And whatever the destination, no one seems to know how to get from here to there.

In the meantime, the companies loom large in the debt markets, just as they did in 2008 when the government took them over. Fannie and Freddie have guaranteed $5 trillion or so of mortgages. The companies’ bonds now reside in banks and insurance companies all over the country, as well as at the Federal Reserve and overseas.

Foreigners held $1.28 trillion of agency debt at the end of the first quarter, Fed data show. That’s down from $1.58 trillion at the peak of the debt bubble in 2007, but still gives U.S. trading partners a substantial stake in the outcome of this debate.

With any luck, Tuesday’s discussions will shed some light on the steps the government might take next, and offer some signs that politicians on both sides of the aisle are taking the issue more seriously. If not, the housing mess will only get more unruly.

“If the debate about mortgage finance’s future cannot be raised above the level of a partisan turkey shoot, the prospects for any meaningful change are likely to be dim,” former Fannie Mae executive Barry Zigas wrote on his blog.