The Brexit process has become almost unspeakably awful. Like a foul smell that nobody really wants to deal with, without action it will only get worse. As party unity collapses in parliament, how we find consensus on the greatest issue of our time will determine what the outcome is – whether that turns out to be a second referendum or the next stage of negotiation. We believe the best path to such a consensus involves setting up a citizens’ assembly.

Just like the Labour party, we represent contrasting communities that nevertheless have much in common. Wigan voted 60%-40% to leave the EU, Walthamstow 60%-40% to remain. One is a northern town fighting to regain the good jobs and dignity that were once commonplace. The other is an inner-city area fighting an epidemic of knife crime and the rise of gentrification. Separated by 200 miles, in both places people are angry, deeply affected by poverty and disempowered by a system that does not respond to their needs.

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Like our constituents, we hold competing views on how to resolve the Brexit stalemate. But we are united in respecting the causes of that fury, and in our belief that only a Labour government can resolve it.

Without a serious attempt to bring our communities together, we will not address any of these concerns. Yet we have lost the ability to listen to one another, and our political discourse has been poisoned by anger and arrogance. Our constituents are not stupid or racist, any more than they belong to “liberal elites”. But politics has become about the box you put someone in to dismiss them, not the bridges you build to find shared solutions.

This shock-jock culture thrives on extremes. People are encouraged to resign from political parties, to antagonise and abuse opponents for effect, for a YouTube hit while wearing a yellow vest, rather than to take the harder path of building support for a different approach.

Throughout history, politics has had to find a path through huge and fundamental disagreements. As President John F Kennedy wrote in Profiles in Courage, it very rarely happens that “all the truth, and all the right, and all the angels are on one side”. Our refusal to acknowledge this is breaking our democracy. As sitting parliamentarians, we feel the disconnect every day, working in expensive, dilapidated buildings where there is fatuous debate about the bumper-sticker on the Speaker’s car while children go without food to eat and mental healthcare when they are in crisis. This isn’t what either of us signed up for.

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This momentous week, and the massive rejection of Theresa May’s Brexit deal by MPs night, will not change this state of affairs.

In the days and weeks ahead, MPs will continue to perform for social media, medicines will be stockpiled, and many more jobs will be tragically lost. There may be a majority to prevent no deal but there is no majority for an outcome – and, crucially, no process that can get us there.

We are nearly three years on from the referendum, and yet we are still arguing about the “will of the people”. Opinion polls and focus groups are conflicted. As has been made abundantly clear, referendums alone, like elections, are blunt instruments that remove complexity in pursuit of simple propositions.

Other countries have seen their democracies strengthened by involving the public alongside parliamentarians in meaningful decision-making. It hasn’t just been in Ireland, where decades of intransigence on abortion and same-sex marriage gave way to a new consensus. It has also happened in Iceland (after the banking crisis), and in Canada and Australia, on issues such as nuclear waste and constitutional reform. President Macron, responding to widespread unrest in France, has recently embraced this approach, understanding its value in reuniting a divided country.

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Like a circuit-breaker, citizens’ assemblies can disrupt the bad habits that have come to characterise Brexit: kicking issues into the long grass, placing party interests over the national interest and assuming the public are unable to cope with hard choices. They don’t replace parliaments or offer politicians an escape from difficult decisions. Instead they reject binary choices – remain or leave, no deal or Norway – and allow randomly selected groups of citizens to explore options in an open forum and make recommendations about priorities to elected MPs, who retain the final say. In a deadlocked parliament, this could be the one approach that could retain the trust of us all that the answers were fair, thoughtful and not predetermined. If, for instance, a citizens’ assembly recommended a referendum, it could also consider what the question should be, providing confidence that there was no attempt by politicians to game the system.

None of this need delay progress. Citizens’ assemblies can be completed in seven weeks, in contrast to the three months of Electoral Commission consultation needed to move to a second referendum. Some might sneer that we already have a citizens’ assembly, in the shape of parliament itself, failing to acknowledge the inability to make progress without meaningful public engagement. Critics might reject the idea for fear of the new. Yet anyone looking at politics right now can see new thinking is urgently required.

This will be another week where Westminster repeats the same tired old conversations. No one is winning, but even as the prospect of no deal approaches, few will be honest that we are stuck. As other nations have found, involving citizens in discussions doesn’t diminish politics or politicians – it enhances the value of the conversation for both. With little evidence that more delay will resolve this situation, the public are desperate for us to change our tune. It’s time we let them help choose the music.

• Lisa Nandy is Labour MP for Wigan; Stella Creasy is Labour MP for Walthamstow