Just a few months ago, the idea of a “people’s vote,’’ or second referendum, on Britain’s withdrawal from the European Union was regarded by most commentators as fanciful. Now, an increasing number see it as the way out of Britain’s Brexit nightmare.

Admittedly, Prime Minister Theresa May is adamantly opposed to the idea, and she insisted once again Monday that there would be no second plebiscite, saying that it would “break faith with the British people.’’

But her options are closing in, and she has changed her mind abruptly before, most recently by postponing the parliamentary vote on her beleaguered Brexit plan, which she says will now take place the week of Jan. 14.

Behind the scenes, some members of her Cabinet are said to be trying to lay the groundwork for a second referendum if — they think when — her Brexit plan is rejected. Some say the fact that May felt the need to reject the idea of another referendum shows how the ground is shifting beneath her.

But there are many questions: Why have a second referendum? How exactly would it work? Would it really settle matters?

When Britons voted to leave the European Union in 2016, they answered a blunt question: Stay or go? But there was little informed discussion of what kind of country Britain would be if it opted for withdrawal. Moreover, Brexit’s proponents portrayed the whole process as an easy win for Britain with no downside.

Boris Johnson, who was later foreign secretary, suggested that Britain could have its cake and eat it; Michael Gove, now the environment secretary, said that in future trade talks Britain held “all the cards.’’ Even after the vote David Davis, the former Brexit secretary, predicted “no downside to Brexit, only a considerable upside.’’

The complex trade-offs that flow from unscrambling more than four decades of European integration were barely explored, and little attention was focused on crucial issues like how to keep open the Irish border. Now that reality has set in, Parliament is paralyzed by the unpalatable choices it faces.

So, the basic argument for another referendum is that voters know much more now than they did in 2016, and that they can therefore make a far better informed decision on the historically critical decision, and in the process break the logjam in Parliament.

Most people agree that May’s plan should be one of the options, since, after all, it is the only detailed plan acceptable to the European Union out there. But then what would be the second choice?

Brexit supporters say the only other choice should be a clean break, a “no-deal Brexit,’’ since voters already decided in 2016 to leave the bloc. But such a rupture would be economically damaging and potentially chaotic, and is the one thing that a clear majority of lawmakers want to avoid. Many of them would like remaining in the bloc to be the alternative.

But why just two options? Why not put all three scenarios — May’s deal, no deal, and remain — on the ballot? Some worry that this risks diluting the vote so that no single choice would get a strong majority, leaving things just as confused as before.

Some have suggested an even more complex vote, perhaps asking two sets of questions: first whether to support May’s deal, and then give those who rejected it a choice between no deal and remaining. This might be done by having two sets of questions on one ballot paper, or perhaps by having votes on two consecutive weeks (like a French presidential election).

Critics dismiss the idea as another incomprehensible complexity in a subject that has already left much of the population confused, bored, or both.

Britain is scheduled to leave the EU on March 29 and would need far more time to set a date and hold a proper campaign. But exit day could be delayed, providing the other 27 nations agree, which they probably would for another vote.

Parliament could approve legislation for a referendum, but it would be hard to see that happening without the government being on board.

Since 2016, Britain has failed to heal the divisions provoked by the Brexit campaign; if anything, they have deepened. Liam Fox, the trade secretary, argues that if remain were to win, people like him would be arguing to vote yet again in a best of three plebiscites. A narrow victory for remaining would set the stage for decades of recriminations over the Great Betrayal, and possibly reignite the fires of right-wing populism.

Yet some say the country’s divisions will persist no matter what the outcome.