Like someone who works with heavy machinery or lives next to a railway line, it’s possible to get so accustomed to the ugly noise that blasts out of our political system that you don’t always hear it.

But whenever I step back, I can tell the sound of our politics has become shriller and angrier, particularly since the referendum of 2016 that split our country in half.

Yet, among the sound and fury, there are the beginnings of a different tone, quietly demanding to be heard. Last Saturday, I went on the march for a people’s vote. It was huge and passionate but there was also a calmness and dignity in this million-strong protest that I have not always experienced in the past few years.

Then, on Wednesday night, my party came together around a position on Brexit that can unite our members, voters, MPs and, yes, the leadership too. I respect the different judgments reached by some of our MPs – none of these decisions is easy – but more than 80% of Labour MPs backed Margaret Beckett’s motion saying that any deal to leave the EU should not be pushed through by parliament unless it has also been approved by the people.

This is a compromise position that provides a clear path to break the deadlock. Whether you voted Leave, like the majority of people in my area did, or whether you voted Remain, the current parliamentary paralysis works for no one.

Our plan recognises that any Brexit deal agreed by this parliament is going to be so different from that promised three years ago – and so potentially costly to our economy and communities, it’s only reasonable to let the people decide whether to accept it. If parliament imposes a Brexit deal on the nation – or decides to scrap Brexit altogether without checking with voters – it would seem undemocratic in the extreme.

We now have until 12 April to either agree a new path to a viable Brexit deal, with an extension to article 50 that means taking part in European elections, or crash out with no deal.

I think Labour should embrace European elections just as we should welcome a general election that might get rid of this broken government. If we go into either contest with a positive policy on Brexit and say any final deal must be put back to the people, I am confident Labour can win. We already have a fund raised by local parties to pay for the campaign and I’ve asked our NEC to develop a plan to get grassroots members involved in selecting candidates.

Ministers have started saying that taking part in EU elections would be somehow be an “affront to democracy”. They no longer make the case for a stronger economy, £350m extra every week for the NHS or amazing new trade deals negotiated by Global Britain. All they have got left in their locker is a promise to give us fewer chances to vote.

The Tories are right to be frightened of elections. Their internal war over Europe, waged for the past 30 years, has led the country to this point. They’ve messed up Brexit that has gone from chaos to national humiliation and are completely incapable of providing the leadership we need.

In contrast, Labour emerged from the past week stronger and more united. Our values have always been internationalist and outward-looking. The biggest gains we made in 2017 were among young people who share those values and want to “build bridges, not walls”. It falls to Labour to develop a modern programme of change to address the root causes of why 52% voted for Brexit.

Whenever a final deal is reached, and whatever that deal eventually looks like, I think this question can only be truly settled by giving the decision to the people. Our plan to put any final Brexit deal back to the people has already begun to unite the Labour party. And, if the last referendum divided our country, I believe a new one might help heal it.

• Tom Watson is the deputy leader of the Labour party