The Brexit deal, as represented by the Trade and Cooperation Agreement (TCA) is the start of a new relationship between the UK and the EU. As the economic and political consequences of Brexit play out, there will be pressure to change the TCA, either in a “softer” or “harder” direction in one particular issue area or another. To try to find out where the EU-UK relationship might be heading in the medium-term we conducted a conjoint survey experiment with a random sample of British voters, where we asked them to choose between different hypothetical “package deals”. This forced respondents to have to make trade-offs across key issues. When faced with such choices, British citizens overall prefer a “softer” form of Brexit to the TCA: where the UK applies EU regulatory standards in return for greater access to the EU’s single market. However, a majority of Leave voters prefer a much “harder” trade-off: of full regulatory sovereignty but continued restrictions on UK exports. We also found that Scottish citizens prefer a closer relationship with the EU across a range of issues than do English citizens. Reconciling these differences will continue to be a challenge for British politicians and policy-makers. This project received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement number 822304. The content of this document represents only the views of the InDivEU consortium and is its sole responsibility. The European Commission does not accept any responsibility for use that may be made of the information it contains.