It's easier if you create a new apple ID which you use for the EU, separate from your existing one, and then login to the phone with your old Apple ID, and login to the app store specifically with the new one.

As a bonus, you can then re-buy all your apps if you want to not have to sometimes toggle between the old / new one to re-download already purchased apps.

Frankly, I'm surprised apple doesn't hard-require you to re-purchase all your apps when you switch regions, it would be a lot more apple-like.