The Senate has already voted to grant President Obama Fast Track authority to negotiate the massive Trans-Pacific Partnership trade deal, but the smart Beltway money says it faces a much, much tougher test in the House. If Fast Track does go down in the House, it could take the whole TPP with it.

And in the House, a big bloc of liberals has hit on what they hope will be a new way to sink Fast Track. It turns on an obscure aspect of this debate that has gone under-covered: A problematic funding mechanism for a program, which would be part of the whole deal, that grants assistance to workers displaced by trade.

In a new letter to Speaker John Boehner and Dem leader Nancy Pelosi, 61 House Dems are demanding a new funding mechanism for the spending for displaced workers, which is called Trade Adjustment Assistance, or TAA. The letter was spearheaded by Congressional Progressive Caucus co-chairs Keith Ellison and Raul Grijalva.

You’d think it doesn’t matter much what 61 House Dems do or say, given that Republicans run the place. But due to the strange, messy process that is coming together to get Fast Track passed, it’s not inconceivable that their opposition could matter. Or, at least, that’s what this bloc of liberals hopes.

Here’s how it all works. House GOP leaders — who support Fast Track and the TPP — are uncertain whether they can get Fast Track passed, because a bunch of House conservatives may vote against giving Obama Fast Track authority even to negotiate a free trade deal, because, well, you know, it’s Obama. As a result, most reports say, Fast Track will need the support of at least 20 House Democrats, or perhaps more, to pass.

There is a group of pro-trade House Democrats who are prepared to vote for Fast Track. But one of that bloc’s leaders, Rep. Ron Kind, flatly told Politico that pro-trade Dems would need Trade Adjustment Assistance to pass first, presumably because pointing to help for displaced workers would make it politically easier to vote to expedite a free trade deal.

The Senate version of Fast Track, in fact, explicitly included TAA to win over pro-trade Senate Dems, thus getting it through the Upper Chamber. But in the House, things are more complicated: Because many House Republicans probably don’t want to vote for federal funding to help workers, GOP leaders are expected to hold separate votes on the TAA and Fast Track measures, the idea being that the first can pass with a lot of Dems, and the second can pass with mostly Republicans.

Now here’s the rub. The Senate version of TAA is funded in part by cuts to Medicare growth.

House liberals don’t like that. In their letter to Boehner and Pelosi, they declare that while TAA is essential, it must not be funded by “cuts to critical social programs like Medicare.”

If all 61 signers of the letter vote against TAA in the end — and all remaining Dems vote for it, which is far from certain — you could need as many as 90 House Republicans to vote for TAA, which is spending to help workers, for it to pass. And if it doesn’t pass, those pro-trade Dems might not vote for Fast Track. Which could doom it in the House, depending on how many Republicans decline to support it. (There’s also another possibility: Depending on how House Republicans structure the rule for this vote, TAA might have to pass in order for Fast Track to pass.)

Pelosi is taking the possibility of a failed TAA vote in the House seriously. A Pelosi aide tells me that she is negotiating with GOP leaders to find a new pay-for to replace the Medicare cuts, since keeping them could end up killing it.

But even if a new pay-for is agreed upon, then there would have to be conference negotiations, to reconcile the House version (with the new pay-for) with the Senate version (with the old pay-for). And Paul Ryan has warned that going to conference is bound to fail and would kill Fast Track.

It’s possible that even if TAA fails in the House, pro-trade House Dems could still end up backing Fast Track. But that, too, would require going to conference, to reconcile the House version (without TAA) with the Senate version (which has TAA). And Ryan says conference will fail.

To be clear, Fast Track could still survive, despite the best laid plots of liberals. Enough House Republicans might vote for TAA for it to pass, whatever the liberals do. And after that, Fast Track could pass with the support of very few Democrats.

But still, liberals are throwing one more obstacle in the way of Obama’s big trade deal. It’s a reminder that there are still plenty of ways that it could go down.