The biggest divide in British politics isn’t between Remainers and Leavers, Blairites and Corbynites or even left and right. It is, I think, the planet-sized gulf between the average Brexiteer in Westminster and the millions of voters who endorsed their cause last summer.

The typical Leave voter is older than the national average, concerned about immigration (particularly the low-skilled kind) and tends to have O-level qualifications at most. Their nominal representatives at Westminster are mostly privately educated, in possession of at least one degree and, in the main, intensely relaxed about immigration.

Above anything else, this was the master­stroke pulled off by Matthew Elliott, the head of Vote Leave, during the referendum campaign: keeping the likes of the Conservative MP Jacob Rees-Mogg (Eton, Oxford) and Daniel Hannan MEP (Marlborough, Oxford) and their lofty talk of sovereignty, trade deals and open borders away from the headlines.

The result was a famous victory, but one that will inevitably end in betrayal. After all, what were the two biggest promises to Leave voters? That the money we send to the EU could instead go to public services – and the NHS in particular – and that we would cut immigration by ending the free movement of European citizens.

However, these outcomes each demand a different kind of Brexit. To have a chance of freeing up more money for the Treasury, Britain would need a soft exit that does not damage the economy; while to end freedom of movement, we must leave the single market, with all the financial repercussions that entails.

Is there another way? What most British people (regardless of their Leave or Remain vote) want, in essence, is for immigration to be under the control of British politicians – without getting poorer.

Although the diplomatic task would be fraught, there is a deal that Theresa May could seek to satisfy these two impulses. First, though, it is important to remember what leverage Britain does and does not have. We cannot credibly threaten to wreak havoc on the European economy without damaging our own, any more than a cyclist can expect a lorry to be intimidated by the thought of a collision. However, we do have two cards to play.

The strongest is our outsized contribution to the EU’s coffers. Nations that are net beneficiaries of European spending have no wish to become net losers, while the creditor countries are understandably averse to being left on the hook for still larger sums of money once the United Kingdom has gone its own way.

The other card is in London’s huge financial services sector, which, in effect, operates as the financial centre of the euro­zone, even though the UK is outside the single currency. And here there is room for a concession: the stress fracture in the EU’s relationship with Britain, even before Brexit, was that the City of London remained outside the regulatory orbit of the European Central Bank.

If Theresa May wanted to unblock the Brexit deal, she could do the following: guarantee that the United Kingdom’s outstanding obligations would be honoured in full. Agree that regulatory controls would be set in Brussels and followed in the UK, which would lose its right to set the rules of European trade and would instead simply have to follow them. May could sweeten the deal by putting the City of London within the reach of the European Central Bank and by continuing to pay into the EU’s coffers.

If the Prime Minister were prepared to offer all of this, she could reasonably expect to ask for an opt-out from the free movement of people without agreeing to a Brexit that destroys the British economy. The sheer cost of such a deal would make it discouraging and impractical for any other European country to follow – easing the EU’s fear of a Brexit contagion – but it would be low as far as the cost to the average British person is concerned.

So what is stopping May? Such a deal would require saying goodbye to the vision of Brexit that Rees-Mogg and Hannan have dreamed of since they were schoolboys. It would mean that the UK would not be able to seek new trade deals but would instead have to slipstream behind the EU, rendering Liam Fox’s department redundant. It would mean shedding the last of Britain’s great power pretensions and accepting a status firmly in the third tier of international politics.

To the average Leave voter, none of this would sound particularly outrageous. It would certainly be more palatable than any of the proposed alternative free trade deals that Britain might conceivably line up. The price for these will range from more immigration (India wants more visas) to unpleasant chlorinated chicken (from the US, where food safety standards are lower).

Yes, this version of Brexit would mean the greatest loss of political sovereignty in British history. It would frustrate politicians on both sides, who would have to endure a new status as observers, not participants, in both European and global politics. Yet it would go some way to making sure that the promises of Brexit are kept.

Who should the Prime Minister betray? The Brexit elite at Westminster, who cling to the idea that leaving the EU will unlock a panoply of trade deals, without any loss to the economy? Or the average Leave voter, who prioritises a cut in immigration and finds talk of tariffs meaningless and dull?

Out of political timidity, Theresa May avoided acknowledging this choice before the general election. Because of her resulting weakness, she must betray the voters, not her party. Disappointment over Brexit will not be felt in parliament, but outside it.