Like any other multi-level democratic polity, the European Union (EU) rests on public approval. However, the nature of the EU’s compound polity – porous boundaries, a weak authoritative binding centre and thin social citizenry bonds – make it vulnerable to existential crises that might trigger disintegration. While public opinion studies in the Eurosceptic and postfunctionalist traditions have recognised these ‘three B’ elements in isolation, our polity-centred Rokkanian approach theorises them as dynamically related, determining fundamental opportunities, constraints and horizons for the future of the EU (Ferrera et al. 2023). We operatise these dimensions with original public opinion data taken from a survey fielded in early 2025 (https://solid-erc.eu/) in 15 EU member states. By intersecting EU citizens’ degrees of satisfaction with the status quo with their likelihood to vote for their country leaving the EU in a hypothetical referendum on EU membership, we identify three main categories of citizens: convinced remainers, who are satisfied and would vote “remain”, convinced exiters, who are unsatisfied and would vote “leave”; and reluctant remainers, who are unsatisfied but still would vote “remain”. We plot their preferences on strengthening the EU polity along different binding, bounding and bonding dimensions, then perform individual-level analysis, locating characteristics associated with these types of citizens, with a particular focus on their view of the perceived political, social and economic impact of EU membership for their country and the EU’s effectiveness in managing past crises. Overall, our work represents a novel attempt at understanding whether Europeans intuitively reason in polity-building terms