The Trade and Cooperation Agreement (TCA) is the start of a new relationship between the UK and the European Union (EU). As the consequences of Brexit unfold, there will be pressure to change the TCA, either in a “softer” or “harder” direction. To determine the potential medium-term direction of the EU–UK relationship, we conducted a conjoint survey experiment with a sample of British voters, where we asked them to choose between different hypothetical package deals. When faced with such choices, British citizens overall mostly support a softer relationship in which the UK applies EU regulatory standards in return for greater access to the single market. However, Leave voters most often support a much harder trade-off of full regulatory sovereignty but continued restrictions on UK exports.