Daniel Leal-Olivas/AFP via Getty Images BREXIT FILES INSIGHT Brits want to have cake and eat it Survey shows Remain and Leave voters have similar Brexit wish list. That’s a problem for Theresa May.

It turns out that Boris Johnson, when he declared that on Brexit he was in favor of “having our cake and eating it,” was in tune with the views of the British people.

Intriguing new research, which claims to be the most in-depth study of what the British want from Brexit, reveals that most Brits — even those who voted Leave — would like to keep the benefits of being in the EU, but not the immigration, thank you very much.

The findings, from the National Centre for Social Research (NatCen) think tank, are in part a vindication of Theresa May’s Brexit strategy. From the off, she made control of immigration her primary red line. That’s an approach that chimes with the British people. A resounding 68 percent want Brexit to mean that EU migrants are treated the same as non-EU migrants — in other words, an end to freedom of movement.

Interestingly, that includes 54 percent of Remain voters.

However, maintaining free trade with the EU is more popular than ending freedom of movement: 91 percent of Remain voters support continuing free trade, and so do a whopping 88 percent of Leave voters.

Other benefits of EU membership are also popular: clean water regulations, restrictions on mobile phone roaming charges, airline delay compensation, university research funding, passporting for financial services — all are supported by large majorities of the British public, including most Leave voters.

It’s hard not to be reminded of the Monty Python sketch: “What has the EU ever done for us? Apart from the free trade, the clean water, the roaming charges etc …”

Of course, the crux of May’s Brexit negotiation will be a trade-off between the popular benefits of EU membership and her wish to exempt the U.K. from the bits that the public doesn’t like. NatCen has done some polling here too. Asked whether they would be willing to accept freedom of movement in order to achieve free trade with the EU, there is a majority (54 percent for versus 44 percent against) in favor of such a trade-off.

Overall, this kind of data is problematic for May. The British public may want to have their cake and eat it but the EU are very clear that this is not possible, and that leaving must come at a cost. The prime minister goes into these negotiations carrying the extremely optimistic hopes of her electorate. There could be trouble for her down the road if she doesn’t come home with the cake.

This insight is from POLITICO’s Brexit Files newsletter, a daily afternoon digest of the best coverage and analysis of Britain’s decision to leave the EU. Read today’s edition or subscribe here.