At every turn, the politics of the Brexit process have been riddled with disastrous illusions: the referendum would settle the issue once and for all; leaving would mean a cash harvest for the NHS; an early general election would give the government a mandate; no deal would be better than a bad deal; as Brexit neared, the country would come together.

It was all wishful thinking, as most illusions are. We are now up to our collective necks in the consequences. Most of the wishfulness was on the side of the leavers and of the two most recent Conservative leaders. Now, though, it is the remainers who have to ask themselves whether they, in their turn, are not falling for another Brexit illusion – this time in the shape of a second referendum, or the so-called People’s Vote.

A significant shift from leave to remain since 2016 among voters in 100 Labour seats could be a game-changer

Theresa May has always brushed aside the second referendum proposal. It is the concession she can never allow. It would mean that Brexit might not mean Brexit after all. It will not happen “in any circumstances”, says Downing Street. But in fact there are circumstances in which it might.

It has always been clear, politically speaking, that only a second referendum could undo the first. It is the only democratic way. That is why May cannot allow that possibility. Yet the advance of the second referendum option up the political agenda is real and inexorable. There is now a clear majority, according to the opinion polls, for a referendum on the terms of a future EU deal. There is also now a majority against Brexit. If the Brexit negotiations get even more bogged down, these majorities can be expected to grow. The argument that the politicians have failed to sort things out, and so the people must decide, will get louder – and will have a certain logic.

One reason for this is that Labour seems to be moving towards embracing the second referendum idea. Keir Starmer, the shadow Brexit secretary, has said it is a possibility. John McDonnell, the shadow chancellor, has refused to rule it out. Now, perhaps crucially, there has been a significant shift from leave to remain since 2016 among voters in 100 Labour leave constituencies, including McDonnell’s. As the polling analyst Peter Kellner pointed out, this reduces the risk of a leave backlash in Labour heartlands. It could be a game-changer.

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What is more, the issue is certain to come up in parliament this autumn. The first likely opportunity will be when MPs debate their “meaningful vote” motion, possibly in November. But there will then be a second opportunity, when parliament scrutinises the promised withdrawal agreement and implementation bill. If the government did a U-turn on May’s opposition to a fresh vote, a second referendum would get a majority, though it would split the Conservatives. Even if the government stuck to its opposition, there might be enough Tory remainer rebels to help carry a second vote amendment.

Let’s assume, though, that Britain decides to hold a second referendum. This is nowhere close to being an end to the matter. In fact there is both an underlying problem – and a whole heap of practical ones. The underlying problem is that Britain has never satisfactorily sorted out whether or how referendums can coexist with our system of representative democracy. The last two years have been an object lesson in the seriousness of this. Last month an independent commission on referendums, commissioned by University College London’s constitution unit, argued that it must be sorted out. But it won’t be. There just isn’t time. The second referendum would trigger a whole set of problems of its own, whatever the result.

Now consider some practical problems. The first of these is time. Brexit is scheduled to take place on 29 March next year. Almost certainly, all 27 remaining EU states would have to agree to extend the article 50 process beyond then. That can’t happen overnight. Parliament would also need time. Referendums need legislation. Legislation can be amended. The bill would certainly not be nodded through. The Electoral Commission and the local authorities that must run the poll need time to draw up rules – and these need to work far better than in 2016. Yet the bill for the 2016 referendum was introduced 13 months before polling day. This vote would have to be in spring 2019. We’re looking at – perhaps – three months max.

The consequences could be as complex and difficult as the consequences of the first vote have proved to be

None of this is impossible, although all of it is difficult. In 2015 Greece managed to conduct a nationwide referendum on its IMF bailout terms a mere eight days after the poll was announced. So the British timetable could doubtless be compressed too. But the more it is compressed, the more the legitimacy issue will be in question. Even the Greek referendum of 2015 had to survive a court challenge.

Then there is the question of the question. What would the referendum ask? The 2019 vote would presumably be a vote on the deal, since that is what People’s Vote campaigners are pressing for. A yes/no vote on the deal would be straightforward if the result were yes, but chaotic to interpret if it were no – with angry leavers and remainers combining to defeat the deal. Maybe that chaos is the price Britain must pay in order to avoid Brexit. But it is to avoid this that Justine Greening, May’s former education secretary, and others have proposed a three-option vote – accept, reject and remain, or crash out – with a preferential voting system. Others have floated the idea of a two-stage referendum.

Britain needs the chance to rescue itself from the decision it took in 2016. We can do so much better than that. So, if this is the only way to stop Brexit in a democratic manner, then let us do it. But let’s not kid ourselves: the whole thing is riddled with risk and uncertainty, the politics are treacherous, and the consequences could be as complex and difficult as the consequences of the first vote have proved to be.

When the time is right, we need a second vote, but without illusions – for we will be messing with an already deeply troubled country.

• Martin Kettle is a Guardian columnist