Critics called it an unprecedented power grab, because the prime minister and her government traditionally control the agenda. But the votes are nonbinding and only express the will of the House of Commons.

“Clearly, it’s incumbent on the government to listen to what the Commons says,” Health Secretary Matt Hancock told the BBC. “But we can’t pre-commit to following whatever they vote for, because they might vote for something that is completely impractical.”

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He added that “the best way through this impasse is the one deal that’s been negotiated with the E.U.”

May, for her part, has objected that the planned votes “overturn the balance of our democratic institutions.”

She noted, too, that the proposals that might be debated — such as a softer Brexit and a Canada-style free trade agreement — have nothing to do with the withdrawal treaty on offer, which is narrowly focused on how Britain leaves the trading bloc and not on the future relationship with Europe. That comes later.

A survey published Tuesday suggested that more than 80 percent of Britons think the government has handled Brexit negotiations badly. It also suggested that 55 percent would now vote to remain in the European Union.

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Britain voted to leave by 52 percent to 48 percent in the June 2016 referendum.

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Rumors are swirling that May could announce her departure date in the hopes of winning over skeptical Brexiteers who would prefer another prime minister to lead the next phrase of Brexit talks.

Or she could call a general election. Or May could plow on — as she is wont to do — and continue trying to win support for her twice-defeated Brexit deal.

But she has clearly lost some control over Brexit machinery.

On Monday, Parliament voted 329 to 302 to assess support for its own Brexit proposals. May’s dwindling authority took a further hit when 30 members of her Conservative Party rebelled to vote against the government, including three ministers who resigned from their posts.

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In a resignation letter published on Tuesday, Alistair Burt, a Foreign Office minister, said, “Parliament should seek urgently to resolve the situation by considering alternatives freely, without the instruction of party whips, and Government should adopt any feasible outcome as its own in order to progress matters. I did not believe the Government was prepared to do that, so had to vote to ensure this happens.”

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This was supposed to be the week that Britain left the European Union. But since that deadline was punted to least April 12, it seems everything is back on the table from Parliament’s perspective.

It was not yet clear which Brexit options would get a vote. The possibilities could include May’s deal, a Norway-style relationship with the E.U., a second referendum, a Canada-style free trade agreement, a customs union with the E.U., a no-deal Brexit and canceling Brexit altogether.

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But there is no guarantee there would be a clear winner.

Some analysts say a Brexit that involved a closer relationship with the E.U. could command a majority in Parliament, but that would be unacceptable to parts of May’s Conservative Party.

It’s possible, however, that the threat of a so-called soft Brexit, or even no Brexit at all, could help to swing the hardcore Brexiteers around to May’s deal.

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Jacob Rees-Mogg, an influential Brexiteer, seemed to suggest he could back May’s deal when he tweeted on Tuesday: “The choice seems to be Mrs May’s deal or no Brexit.”

May needs the support of the Brexiteers in her own Conservative party; the Democratic Unionist Party, the small Northern Irish party that props up her government; and probably a number of rebel Labour lawmakers. Currently, the odds are not in her favor.

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The DUP showed no signs of budging on Tuesday. Writing in the Daily Telegraph, DUP Brexit spokesman Sammy Wilson said a one-year extension would be preferable to May’s withdrawal agreement, which he claimed doesn’t deliver Brexit.