LONDON—More than two years after Britons voted to leave the European Union, the U.K.’s politicians still can’t agree on what kind of Brexit they want. Now more lawmakers are calling for a do-over, a second referendum that could potentially reverse the results of the first.
About 150 members of Parliament back a new vote, including Labour, Scottish nationalists and a handful of Conservative legislators, according to Dominic Grieve, a Conservative member of Parliament who is supporting a fresh vote.
That is still far from the roughly 320 needed to pass a measure to hold another referendum.
Moreover, it isn’t clear what Britons would vote on. A ballot could ask if they want to reverse the 2016 decision to leave the EU. Alternatively, Britons could be asked if they support a deal Prime Minister Theresa May has negotiated laying out the terms on which the U.K. leaves the EU or if they would prefer to leave without any negotiated deal.
Breaking the Stalemate
Chances are rising that the U.K. Parliament could opt for a second referendum as a way out the Brexit impasse. Here are the options.

Extension needed
No extension
Negotiate future
relationship
Passes
Transition period
Fails
Government loses no
confidence vote
General election
May calls election
New proposal
May’s deal
Second referendum*
No Brexit
No deal
Parliament votes down
every option
Start over

Extension needed
No extension
Negotiate future
relationship
Passes
Transition period
Fails
Government loses no
confidence vote
General election
May calls election
New proposal
May’s deal
Second referendum*
No Brexit
No deal
Parliament votes down
every option
Start over

Extension needed
No extension
Negotiate future
relationship
Passes
Transition period
Fails
Government loses no
confidence vote
General election
May calls election
New proposal
May’s deal
Second referendum*
No Brexit
No deal
Parliament votes down
every option
Start over

No extension
Extension needed
Negotiate
future
relationship
Transition
period
Passes
Fails
Government
loses no
confidence vote
General
election
May calls
election
New
proposal
May’s deal
Second
referendum*
No Brexit
No deal
Parliament
votes down
every option
Start over
*A two or three option ballot is possible
Source: Staff reports
But more and more lawmakers believe a referendum may be needed to break a deadlock in Parliament, which appears unable to agree on any Brexit divorce agreement.
Prime Minister Theresa May has repeatedly rejected the idea of proposing another referendum on Europe, saying it would be highly damaging for British democracy. “This risks dividing the country again,” she told Parliament on Monday. “We should be striving to bring it back together.”
Those advocating a fresh ballot say Britons didn’t know what Brexit really entailed when they voted to leave the EU in 2016. Now that the exit terms are clearer they should be asked to vote again.
“This is not the Brexit that was promised,” says Jo Johnson, a Conservative lawmaker who recently quit as a government minister in protest of the deal to get the U.K. out of the EU. “That is why we need another vote.”
Mr. Johnson wants to remain in the EU. His older brother and former Foreign Secretary Boris Johnson is a prominent advocate of the campaign for the U.K. to break with the bloc.
Britons voted 52% to 48% to leave the EU in 2016. Currently, 53% support staying in the EU, according to an average of six recent opinion polls compiled by website What UK Thinks. Pollsters, though, caution against concluding there has been a definite switch in public opinion, since neither side has definitively pulled away and most surveys have a margin of error of 3 to 4 percentage points. A recent ComRes poll put public support for another Brexit vote at 40%.
Mrs. May, who is battling to wring more concessions from Brussels on the terms of Britain’s withdrawal, will submit her bill to lawmakers in mid-January.
If lawmakers run out of Brexit options, they will have to back another referendum to avoid crashing out of the EU in March without any deal that governs the terms of its economic relationship with its biggest trading partner, says former British Prime Minister Tony Blair. “When you exclude the impossible, what remains, however improbable, is the answer,” he says.
The once-marginal campaign for a “People’s Vote” has attracted celebrity support, including Harry Potter author J.K. Rowling and three out of the four living former British prime ministers, John Major, Mr. Blair and Gordon Brown. (David Cameron hasn’t stated his view.)
“This is becoming more of a potential outcome,” says Mr. Grieve.
But the path to a second referendum is complex. If Mrs. May’s deal is rejected, the government could call for a referendum to push it through. But for the government, that would be a last resort.
Alternatively, an amendment requiring another referendum could also be attached to subsequent Brexit-related legislation. The final scenario: if Parliament is still deadlocked after the January vote on Mrs. May’s Brexit bill, a coalition of lawmakers could band together to call for a fresh referendum to avoid leaving the EU without a deal.
That a quorum of British lawmakers could unravel Brexit isn’t unimaginable. Nearly three-quarters of parliamentarians backed remaining in the EU during the 2016 referendum.
Whatever route is taken, its probability for success would increase if the main opposition Labour Party, which has been wavering, backed it.
Experts at University College London estimate it would take 22 weeks to organize another referendum, though some lawmakers say it could be done more quickly. European officials said other EU governments would likely agree to postpone Britain’s scheduled EU exit date on March 29 to allow one to take place.
There are three possible options for the ballot-paper question: Mrs. May’s deal, no deal and no Brexit.
No British referendum has contained more than two options, so it isn’t guaranteed the ballot would include an option to remain in the EU.
Nor is it evident that people wouldn’t make the same choice as in 2016: to leave the bloc. “At least it will be a better-informed decision,” says the younger Mr. Johnson.
Write to Max Colchester at max.colchester@wsj.com and Jason Douglas at jason.douglas@wsj.com
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