I spent a bit over a week in Italy, and got asked for a green pass (Italy's vax passport) in three types of locations: McDonald's, Starbucks, and the dining are at Rome airport.
The latter was particularly funny as you're still allowed to eat without one. You just have to stand up at tables provided for that purpose, located in tbe same area.
In all three cases the staff said _only_ green pass was accepted. _Not_ proof of vaccination from non-EU countries. Which is supposed to be accepted!