You mean like they've been doing since the 1950s with the the largest physics laboratory in the world (CERN)? Or more to the point, the European High-Performance Computing Joint Undertaking (EuroHPC JU) of which 27/27 EU member states participate?
Even at the smaller level the ethos and ambition is clear - take EURO-3C: the Horizon Europe project aimed at delivering a pan-European sovereign infrastructure that integrates Telco, Edge, Cloud, and AI capabilities under a federated model which has 70+ Euro-participants.
I think that is the way forward: work with whoever has common interests and is willing to work together.
I think the point is that the EU does not necessarily make cooperation between governments any easier.