The Brexit debate is coming to a head at a moment when Britain’s political parties seem to be coming apart. Maybe that is not surprising. The 2016 referendum usurped the role of parliament and MPs. And both Labour and Conservative parties are being pulled by the far right and far left. The mess of Brexit is the result. Millions of voters are left feeling not only frustrated but homeless, with unprecedented strain on the stability and legitimacy of the two-party system.

The defections from the main parties last month were not just about Brexit. Labour’s antisemitism crisis also reflects dangerous wider trends. It is horrifying to see the Jewish roots of MPs turned into a weapon against their political views; to see sectarianism so deep that reason is lost; to see the anonymity of social media toxify political debate.

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But while the question of Labour’s future will take some time to resolve, the Brexit question is now down to hours. The prime minister is trapped in what can only be called frenetic paralysis, desperately trying to move forward but unable to do so because of the competing pressures on her. Meanwhile the country heads for the cliff edge.

There is only one way to safeguard the political legitimacy, never mind the economic stability, of the country, and that is to re-engage the people. And here is the madness of the current situation: the arguments made against a second referendum apply, in spades, to the prime minister’s deal. It is said that a referendum would be divisive, prolong the Brexit agony, and fuel the far right. But all of these arguments can be made about the prime minister’s deal.

It is true that a further referendum will require people to take sides. But we are divided already, because the 2016 referendum made a mockery of the principle of informed consent, and Theresa May’s deal satisfies no one. Leavers say it is not what they voted for. Remainers say their opinion was not properly taken into account. And the choices, as the recent Institute for Fiscal Studies report made clear, are only going to get harder, as the reality of Brexit works its way through the economy.

Facebook Twitter Pinterest ‘I deplored David Cameron’s argument that he had no option but to put a referendum in his 2015 manifesto.’ Photograph: Leon Neal/PA

The idea that the May deal will end the Brexit trauma is also misplaced. This deal is only the beginning of the negotiations, and in all likelihood the easiest part. The trade-offs associated with the choice of our Brexit future – between alignment with European regulation and the exercise of a new-found freedom to deviate – are only going to get harder.

The argument that a second referendum would boost the far right has been presented in cynical terms by the transport secretary, Chris Grayling, and in an honourable and serious way by various Labour figures. But the counter case is that the politics of anger represented by nativists and nationalists, on the mainstream right as well as the extreme right, is on the march anyway, and finds vent already. While a referendum would give a platform to Nigel Farage, it is not as if he lacks a platform without a referendum.

Ploughing on, when we know that the painful choices that have been avoided by May and Jeremy Corbyn for more than two years will only multiply, is not a salve; it is itself a recipe for more division. May’s bad deal is only an indicator of more pain to come.

A further referendum is not just a way for voters to decide whether to incur the economic and social cost of a final Brexit deal. That would be a good reason – the metaphor of putting down an offer on a house but waiting for the surveyor’s report before deciding whether to sign the contract makes the point. Leading Brexiteers, from Jacob Rees-Mogg to David Davis, previously argued for two referendums as “bookends” of the democratic process. There is nothing undemocratic about giving voters the final say.

A referendum is also a way for the country to renew its democracy. Citizens’ assemblies, local debates, new mechanisms of popular engagement, new focus from broadcasters and social media companies on fact-checking and fake news, new protections against Russian interference – they will all face catcalls that they are high-minded or boring. But they have been shown in other contexts to overcome the danger that it is only those shouting loudest who get heard.

There is a danger that the demagogues take over again in the course of another referendum campaign, but only if we neglect the lessons about how to engage voters in a serious way. The evidence from Ireland – where two referendums covering gay rights and abortion, both core to deeply held views of national identity, served to bring closure and heal divisions rather than allowing them to fester – shows that referendums don’t have to be divisive.

Amid the dangers ahead, the proposal from MPs Peter Kyle and Phil Wilson shines through the gloom. Pass the May deal and put it to a vote. Then there can be a proper debate and a proper answer.

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This will require time, which is not on our side – as a result of May’s incredibly unwise decision to trigger article 50 in March 2017, before there was a negotiating position on the British side. The lesson of the past two years is not to rush further, but to use time wisely and sequence decisions sensibly.

I didn’t expect to be arguing this case. I spent three years as foreign secretary arguing against Tory demands for a referendum on Europe. I deplored David Cameron’s argument that he had no option but to put a referendum in his 2015 manifesto, and Labour’s decision after the election of that year to vote for the referendum in parliament. But paradoxically I don’t see any way out of the current malaise without doubling down on popular participation in a national vote. The Brexit that people were promised in 2016 is not available, but Brexit cannot be cancelled without popular say-so.

As MPs try to find a way through the Brexit maze, it is clear the country has driven into a dead end. It is going to take creativity and flexibility to get out of it in a way that unites people for the huge challenges of reform and renewal that lie ahead. A referendum should be part of that process – and for positive reasons, not just the lack of a majority for anything else.

• David Miliband was the UK’s foreign secretary between 2007 and 2010