Parliament is about to enter another decisive week in the Brexit deliberations. On Tuesday MPs will vote on Theresa May’s deal, and, if it falls, on a no-deal Brexit and the question of extending article 50. Yet little has changed since the last time MPs were asked to vote. The deal is essentially the same as before and likely to be rejected once more, yet there appears no clear majority in parliament for any alternative.

Something fundamental has to change. The Labour leadership, having seen its alternative soft Brexit proposals defeated, has formally announced support for a second referendum, suggesting this to be the logical consequence of the formal party conference position established last autumn. Yet this position, while popular among party members, is fraught with danger if handled badly.

In general, those who oppose a second referendum remain suspicious of the motives behind the people’s vote campaign

The British public are deeply divided over a second referendum. Unsurprisingly, most who voted Leave are against, while those who supported Remain tend to be in favour. In general, those who oppose a second referendum, including large numbers of Labour MPs, remain suspicious of the motives behind the People’s Vote campaign. Essentially they view it as a vehicle to overturn the 2016 referendum result; an outcome the campaigners never accepted. As such, many argue, it will further erode trust in the political system and aggressively fuel the far right.

But does a people’s vote have to be as divisive as many fear? Could it provide a democratic route through the present Brexit impasse? Might handing the British public the final say on a topic politicians have proved incapable of resolving give popular legitimacy to any final Brexit decision, especially when compared with an unpopular deal cobbled together under a last-minute cloud of buy-offs and arm-twisting? The devil is in the detail.

There is no doubt that the British public are angered by the conduct of the Brexit deliberations. Recent polling from Hope not Hate has found that only 2% of the public have been impressed with the way MPs have handled Brexit. More than half the country, 55%, think our political system is broken, with only 34% saying it is working albeit not perfectly. Two thirds of people (68%) say there is not a political party that speaks for them, up from 61% last July.

Part of the public’s frustration is being shut out of the process. Poll after poll has shown that people want to be consulted and involved in the process. The parliamentary impasse has only deepened this frustration. Could a new referendum help resolve these tensions through involving the people in the final decision? Advocates for a people’s vote argue for it on these simple democratic grounds.

Yet a new referendum comes with huge risks. The Labour leadership went beyond the party conference position when it recently announced that a second vote would be a straight choice between May’s deal and remaining in the EU. This would be both extremely dangerous and fundamentally wrong. If anything would create a rightwing backlash, this specific approach to a second referendum would be it.

Keeping no deal off the ballot would lead to understandable claims that the vote was being rigged

Keeping no deal off the ballot would disenfranchise 30%-40% of the population and lead to understandable claims that the vote was being rigged. For a second referendum to be legitimate it has to be considered fair. All sides need to feel that they have a chance of winning – all views need to have skin in the game. If not, the process will be seen to have been corrupted.

The former Ukip leader Nigel Farage has already said that he will call for a boycott of any referendum that excludes no deal. Many will follow his lead. The referendum will immediately lose its legitimacy, especially if Remain wins, and the anger will be on a scale we have not yet experienced.

Giving the public the final say on how, or even if, we leave the European Union might prove the most democratic way of breaking the impasse. It might also prove a popular choice, with more than 60% in four separate polls backing the public having the final say in the event that parliament cannot decide.

While a new vote would reopen wounds, it could offer a degree of public consent and legitimacy for the final Brexit decision – but only if seen as part of a legitimate, democratic process. The danger is that the current approach adopted by both the Labour leadership and advocates of a people’s vote will achieve precisely the opposite.

• Jon Cruddas is Labour MP for Dagenham and Rainham; Nick Lowles is director of Hope not Hate