On Saturday I joined 700,000 people on the streets of London to demand a people’s vote on Brexit. The atmosphere was vibrant, friendly and politely furious. From all over the country people came, anxious to make their anger, despair and incredulity known. It was fun and important. The mainstream has been abandoned on Brexit by our political class and there is a catharsis to taking to the streets and demanding that our politicians listen.

I have to confess, street politics are not my usual style. I have been called many things in public life, but the cap that best fits is that of the centrist dad. In my career and in my jobs in government I have tended to look for the solution and to try to avoid the ideology. I began the academies programme because I wanted to do what worked for disadvantaged kids, not cling to models that looked good and delivered little. I drove HS2 through parliament and worked for cross-party support because I knew that productivity and inequality in this country could only be tackled by massive long-term infrastructure investment. Despite my deep misgivings about austerity and the harm it would do, I agreed to chair the national infrastructure commission under a Tory government, because I believed that delivering infrastructure investment could help build a brighter future for businesses and families. I am a pragmatist. I do what works.

I marched for a referendum because the last referendum has left me – and our country – with no choice

So why, then, in middle age have I picked up a placard and taken to the streets? Why am I protesting rather than participating? Why, like hundreds of thousands of otherwise sensible and moderate people, was I in Parliament Square on a sunny afternoon cheering my lungs out at Delia Smith? Well, because there simply is no way to make this Brexit thing work.

In the end, pragmatism requires a workable compromise. But none exists on Brexit. When we voted in 2016 we were offered not two alternative destinies but thousands of parallel futures. The two leave campaigns contained within them a violent disorder of multiple personalities. Boris Johnson claimed we could remain in the single market and leave the EU – the origin and definition of “cakeism” as a political style. Nigel Farage warned of imminent Turkish accession while insisting that leave was all about getting our old passports back and exciting new trade deals around the world. Jacob Rees-Mogg reassured his friends in the City that post-Brexit Britain would be a haven for them, a sort of Singapore-on-Sea, while “lexiteers” like Kate Hoey promised us a massive programme of state aid. Had this been a general election, with leave as a party proposing to govern, this mishmash of pipe dreams and contradictory fantasies would have been laughed out of town. In a referendum, it got away with it.

Two and a half years on, the Brexiteers still can’t agree on what Brexit actually means. They are tearing each other apart – over “Chequers”, “Canada-plus-plus”, Efta and everything in between – and in the process they are tearing the country apart. Why? Because no Brexit deal can deliver to British voters on the inconsistent but alluring mess that they were promised in 2016.

That’s why I marched on Saturday. That’s why this centrist dad – alongside hundreds of thousands of other marchers – took to the streets to demand a people’s vote. Not because I am an enthusiast for referendums but because a referendum on what we know is the only way to overcome the problems created by a referendum that was won by promising the impossible. I marched for a referendum because the last referendum has left me – and our country – with no choice.

Angry as I am, the pragmatist remains. No amount of chanting or of marching can extinguish in me that most centrist-dad of characteristics: the desire to actually fix the thing rather than merely to protest against it.

And so I have to say this: the march was a wonderful, impactful moment that showed the political class we are serious – but it is just the first step. What is coming next is not a battle of public will but of parliamentary pressure – we face a Commons crisis the likes of which we have not seen since 1940. Now, as then, MPs have the choice between working across party lines to achieve what is right – Churchill as PM, a referendum on Europe, or doing the easy thing – Chamberlain continues, Britain drifts out of the EU. The moment is historic and each of our representatives in parliament should understand that the choice they make on this vital matter will determine how they are remembered. Were they brave, wise and patriotic? Or did they simply throw their hands up, mumble something about how it was all “very complicated” and vote to make their constituents poorer and less safe without giving them a say?

That is the sure and certain knowledge our MPs must take into the lobbies when a vote comes – as it will – on giving the British people the final say. History and their constituents will judge them fiercely if they try to silence us. So I say to my fellow centrist mums and dads, sons and daughters – write to your MP now, today. Tell them you are watching and history is too. Demand your say. The march let them see we are here. Now we need them to actually listen. We need sensible people to stop being quiet and to plod – pragmatically, passionately – on.

• Andrew Adonis is a Labour peer and former minister