LIVERPOOL, England — Members of the opposition Labour Party on Tuesday voted overwhelmingly to support a second referendum — “a people’s vote” — on Brexit.

But what kind of vote, exactly, and when?

The most cathartic moment of the annual party conference came Tuesday morning, when Labour’s Brexit secretary, Keir Starmer, improvised on his prepared remarks and stated, “Nobody is ruling out remain as an option.”

The audience of several thousand burst into applause and stood for long minutes. Members told The Washington Post they were hopeful that somehow there was a chance that Britain’s exit from the European Union could be reversed.

“I’m desperate to remain — for my children,” said Tony O’Malley, 61, a retired lawyer and father of three. “It doesn’t matter if I can go to study in Berlin, but I’d like them to be able to.”

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The enthusiasm in the hall was in line with recent polls showing that the vast majority of Labour Party members, especially its young activists, want to remain in the E.U.

Yet there is little support from most of the party leadership for a complete do-over, as opposed to a thumbs-up or -down vote on Prime Minister Theresa May’s final proposal for the relationship between Britain and the E.U. And because no one can predict what will happen with the pending divorce, the measure passed by dues-paying Labour delegates was necessarily couched in hypotheticals.

Should the British Parliament vote down whatever deal May manages to strike with the E.U. in the next two months — a distinct possibility, since she faces opposition from both Labour and hard-liners within her own Conservative Party — or should the Brexit talks in Brussels implode and there be no deal at all, Labour believes “the best outcome would be immediate general election,” an election that the hard-left Labour leader Jeremy Corbyn thinks he can win and current polls say could be tight.

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The Labour motion continued, “If we cannot get a general election, Labour must support all options remaining on the table, including campaigning for a public vote.”

Corbyn stressed that there should be a vote to approve or oppose any deal May might strike with Brussels. But he and other party leaders stopped short of endorsing another possible referendum on the essential question of whether to leave or remain.

Emily Thornberry, who serves as Labour’s shadow foreign secretary, warned that a replay would be “deeply anti-democratic.”

Brendan Chilton, a leader of the group called Labour Leave, called it “a betrayal of the very highest order” against the millions of Labour supporters who voted in June 2016 for Britain to leave the E.U.

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Gareth Snell, a Labour member of Parliament whose constituency voted 70 percent to leave the E.U., said the promise of a do-over referendum had “more fudge that a Cornish sweetshop.”

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He said the sequence of events that would result in a new referendum seemed unlikely and that the focus instead should be on the underlying forces behind Brexit.

“We’ve come up with a technical fudge that means we have standing ovations in the conference hall and the delegates can go back and feel better about themselves,” he said. “The membership and the electorate in some parts of the country are very different beasts, and, if we are not careful, we run the risk of ignoring parts of this country that used the Brexit referendum as a cry for help. And we dismiss that cry for help at our peril.”

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Among activists at the conference, though, there was a different cry.

Will Harrison, 58, works in technology and was handing out yellow stickers that read “Bollocks to Brexit.”

He said Labour was “moving in the right direction,” but that the option to remain in the E.U. needed to be an option in any future vote.

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“If it was just a referendum on whatever Theresa May brings back, or a cliff-edge, catastrophic Brexit, that’s not really much of a choice,” Harrison said. “I think if we have another referendum, it has to be a serious referendum with the option to remain.”

Steve Gavin, 52, a civil and structural asset manager who was wearing a T-shirt that read “Liverpool for Europe,” said Corbyn and other Labour leaders are “old-fashioned, socialist Euroskeptics” who are “looking at the small proportion of the electorate that would resent change in policy,” as opposed to the “massive amount of modern middle-ground voters that they would pick up with a change of policy. They took those for granted in the last election; they cannot afford to take those moderates for granted in the next election.”

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He said the country was shifting toward a pro-remain position in part because reality was sinking in. “People are now staring it in the face, the horror stories of a couple of years ago: Planes will be grounded, we can’t get goods into the ports, we can’t feed ourselves. All of a sudden, it’s very real.”

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Brenda Bixter, a 50-something musician and teacher, said she felt “bereaved” after the Brexit vote and hoped the conference would be “pivotal.”

“Anything can be stopped,” she said.

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