This concern has deep roots in our history. The Cod Wars with Britain were fought precisely over the principle that we control access to our own waters. It's natural to worry that EU membership would undo that.
The short answer is that the Common Fisheries Policy allocates quotas to member states, not to foreign fleets. Each country receives a fixed percentage of the total allowable catch for each stock, and it's up to that country to decide how to distribute its quota domestically. Spain doesn't get to send trawlers into French waters just because both countries are in the EU — France manages its own zone and allocates its own quota to French vessels.
There is a historical layer called "equal access to waters" in the CFP, which in principle allows EU vessels to fish in any EU waters. But in practice, this is heavily restricted. Coastal states manage their waters out to 12 nautical miles exclusively, and the broader 200-mile zone is managed through the quota system. The actual pattern across the EU is that member states' own fleets fish their own waters, using their own quota allocations.
That said, there are mechanisms for quota swaps and transfers between member states, and some countries do trade access arrangements. We'd need to be alert to pressure for such arrangements during accession negotiations. But nothing in the CFP would force us to open our waters to foreign fleets against our will. The principle of sovereign management within the EEZ, combined with the relative stability quota system, provides a robust framework — as long as the accession terms are negotiated carefully.
Sources: EU Regulation 1380/2013, Art. 5 and 16–17; European Commission, "CFP — Access to Waters" factsheet 2023.