Half in, half out?
Freedom of movement, access to healthcare abroad, voting rights - some fundamental aspects of British life in the EU must be clarified before Brexit happens.
A Luxembourg liberal MEP, Charles Goerens, has proposed offering British citizens the option of retaining their EU citizenship for a fee. This "associate EU citizenship" idea could be part of the Brexit negotiations but it raises all sorts of legal questions.
The need for treaty change makes this initiative very hard to achieve by the likely 2019 deadline for Brexit.
It could only become law after a treaty change because it would change the nature of EU citizenship.
But aggrieved pro-EU Britons may welcome the fact that Belgian MEP Guy Verhofstadt has backed the proposal.
Mr Verhofstadt, an influential liberal, will be the European Parliament's lead negotiator on Brexit.
EU leaders will inevitably have to change the treaties in the next few years because the UK's role will have to be deleted or amended in EU texts.
However, a change to the nature of EU citizenship would require full ratification by all 28 member states - and that could not be done quickly, Prof Catherine Barnard, an expert on EU law, told the BBC.
But the proposal, she said, at least showed a willingness in the EU to "find creative ways to help the 48% who voted to Remain [in the EU]".
Camino Mortera-Martinez, an EU justice expert at the Centre for European Reform, said there was "no appetite for treaty change in Brussels at the moment".
Next year politicians in the Netherlands, France and Germany will focus on general elections. They will want to avoid EU treaty changes, which are notoriously time-consuming and difficult. It took nearly nine years to draft and enact the Lisbon Treaty.
Empowering MEPs to continue representing some UK citizens after Brexit would be another big legal hurdle.
Prof Barnard questioned how an MEP could represent an area where "half the constituents are not even associate EU citizens". "That is a non-starter," she said.
If British "associate EU citizens" were to have continued freedom of movement, the UK would have to offer the EU something in return, both experts argue.
But that is very problematic. The Brexit vote on 23 June made curbing immigration from the EU a top priority for Mrs May's government. Freedom of movement will be one of the thorniest issues.
It would also be hard to get agreement on a fee for EU citizenship, Prof Barnard said. For example Spain, hosting many elderly Britons who use its healthcare system, might demand a high fee.
If it took the form of a new EU tax, it would require extra bureaucrats to collect and allocate the income.
The European Court of Justice would have to oversee associate citizenship - another thorny issue, because the whole idea of Brexit is for the UK to "take back control". So how could an EU court retain jurisdiction over UK citizens?
And it could be discriminatory to offer associate EU citizenship only to UK citizens. Many Serbs and Turks might also demand it, as they would like to join the EU.
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