If there were another Brexit referendum, who should lead the campaign to stay within the European Union? Pop this question to a prominent politician on the Remain side and you sometimes get the response, usually after a faux modest pause, that maybe the best face of the cause would be the person that you are looking at.

Ask a non-politician who would maximise the chances of reversing Brexit and you will often get the answer: “Anyone but a politician.” Speak to some of those involved in the People’s Vote campaign and they will reply that it wouldn’t necessarily require a single, dominant personality. The Brexiters achieved their narrow victory in the June 2016 referendum without a leader. The campaign to overturn that result looks more appealing, more grassroots and less vulnerable to the charge that it is an “establishment” endeavour and better capable of speaking to people on all points of the spectrum, because it speaks with diverse voices.

There are not always obviously correct answers to these questions. Famous political faces, whether they be a Tony Blair or a John Major, both of whom would be on the anti-Brexit side of another referendum, come dragging loads of baggage. Keeping politicians on the sidelines of a campaign always sounds attractive – until you try to front it with business people or celebrities and they collapse on contact with their first searching interrogation. Assembling a wide cast of campaigners sounds like a good idea, but won’t be if it leads to a rudderless effort accompanied by an incoherent cacophony of messages. How a campaign should be fronted is just one of the issues – and not even the most important – facing those who want to stop Brexit.

The Remain side did suffer from looking like the Establishment party in 2016, but those roles have since reversed

They are on a roll at the moment. For a long time, the idea of another referendum looked like a fringe obsession without much backing even among those who voted Remain in 2016. Support has been growing among the public and it is now treated as a plausible scenario in the likely event that there is total deadlock in parliament. This is tribute to the vim and dedication of the People’s Vote campaign, an umbrella for nine organisations. It is also testimony to the chaos unleashed by the Brexiters in the 30 months that have passed since the original referendum. They are falling out and falling apart, collapsing under the weight of their undeliverable promises and irreconcilable contradictions. By castigating Theresa May for coming up with a deal that is worse than continued EU membership, some of the hard Brexiters are helping to make the case for giving the nation the opportunity to think again. By contrast, the anti-Brexiters have a unity of purpose and the sense of vindication that comes from seeing the implosion of their opponents’ enterprise.

The Remain side did suffer from looking like the Establishment party in 2016, but those roles have since reversed. The young and student-led anti-Brexit campaign group FFS, which, obviously, means For our Future’s Sake, has the energy and fun of an insurgency. Brexiters are exposing their fear of another referendum by snarling that it would be “an establishment coup”. This rings very hollow. It was David Cameron (old Etonian) who plunged Britain into this mess in a vain attempt to manage his party’s divisions over Europe. The chief snake-oil salesman of the Brexit campaign was Boris Johnson (OE). The Captain Mainwaring of the hard Brexit platoon in parliament is Jacob Rees-Mogg (OE). You can’t get much more elitist than that Eton mess. There is nothing undemocratic about offering the people the chance to reconsider their decision now that they are in fuller possessions of the facts about what that privileged trio have inflicted on their country.

So the prospects of securing a people’s vote, while still not certain, are looking a lot more promising than at any time previously. Which means anti-Brexiters now have to do some hard thinking. There is an understandable immediate focus on achieving another referendum, but that must not be at the expense of planning for what will have to happen for it to be won.

One of the biggest questions is about the concentration of effort. Should the primary goal be to maximise the mobilisation of those who are already sympathetic to the cause? Or should they be paying at least as much attention to winning the hearts and changing the minds of people who voted for Brexit the first time around?

Most of the focus of the People’s Vote campaign to date has been devoted to galvanising opposition to Brexit and channelling its anger into rallies, marches and lobbying of parliamentarians. One of the unanticipated side-effects of Brexit has been to get people to demonstrate their commitment to the EU and in large numbers. That is a remarkable change to the atmosphere of 30 months ago. During the original referendum, there were very few people willing to make a positive case for membership of the EU. David Cameron was particularly poorly equipped to do so. Having been a party to 20 years of relentless rubbishing of Europe, which never acknowledged the contribution made to the EU by Britain or the benefits that this country derived from membership, he could not credibly pivot into an enthusiast. In the summer of 2016, the worst were full of passionate intensity and the best lacked all conviction.

That is no longer the case. Conviction is no longer the exclusive preserve of the Brexiters. If anything, more of the passion is now to be found among their opponents. You can even get 700,000 people on the streets to march in favour of EU membership, a sight never previously witnessed in British politics. It is conceivable that a referendum could be won and Brexit reversed on the basis of maximising the turnout among the already converted. Combined with changes in the the electorate since 2016 – elderly Brexit supporters are steadily dying to be replaced on the electoral roll by younger, anti-Brexit voters – that might be sufficient to win.

It is never a smart idea to tell voters that they have been fools. Even, perhaps especially, when they have been

This, though, would be a highly risky route to victory. It might not be enough to win. It might be enough to win, but only very narrowly. Brexiters would instantly cry “cheat” or “best of three then”. Many politicians who might be expected to incline to another referendum are highly wary because they fear that it would inflame the divide exposed by the first. On the one side, younger, more metropolitan, more internationalist, more affluent and better educated voters; on the other, older, more socially conservative and “left behind” voters.

This split is real and serious, but it is not a reason to deny the country another referendum. Otherwise, we’d not have elections either, on the grounds that they can be divisive. It is cause to be concerned that another referendum would widen the chasm between two antagonistic, and often mutually uncomprehending, tribes of Britons.

So it won’t be good enough to have an anti-Brexit campaign that speaks only to those already inclined to be sympathetic. The tougher but equally essential task will be engaging with those who voted Brexit and addressing the deep grievances that led them to do so. Here the anti-Brexiters face a difficulty. It is moot how much you can plausibly promise about tackling economic frustrations and improving life chances when you are a campaign, not a government. What the anti-Brexiters can – must – do is show that they have heard the anger of the alienated and understand the causes of their discontent in a way that the Cameron campaign of 2016 never even tried.

We know that some of these voters are open to persuasion that Brexit is not the answer to their anger. There’s evidence in the polls of Brexit voters suffering buyers’ remorse after 30 months of long and painful education about what it actually entails and the exposure of the Brexiters’ empty promises. Here again, though, anti-Brexit campaigners have to be extremely careful. It is one thing to hold Boris Johnson and his gang to account for the mendacities they peddled in 2016. It is another thing to say that the millions of voters who bought into the bogus Brexit prospectus were idiots and suckers. It is never a smart idea to tell voters that they have been fools. Even, perhaps especially, when they have been. The anti-Brexit campaign has to show respect for Leave voters and for the reasons that they voted Leave. They may be willing to change their minds but they won’t hear the arguments for doing so unless they are convinced that they have been listened to as well.

For anti-Brexiters, preaching to the choir is the easy bit. The big challenge will be engaging the unconverted.

• Andrew Rawnsley is an Observer columnist