I am a dual US/ITA citizen and I recently entered the EU with my US passport and got a 90 day visa. I decided I want to stay longer than 90 days and I have my EU passport with me. I am wondering if that will cause issues when I decide to return to the US or trigger any other issues during passport controls when leaving the EU if I overstay 90 days. I've read it isn't possible to overstay if you are an EU citizen, but it would look odd on your (US) passport to not have an exit stamp. Also when I leave the EU to go back to the US I would be using my US passport, so not sure how to handle that with it showing an overstay.
Can/should I go to Gibraltar and get an exit stamp, then re-enter with my EU passport? Or is this not necessary and can I just stay longer than 90 days, then show my EU passport when anyone asks about my overstay?