I believe this is the expected behavior. If you are overloading, you have to provide all of the overloads, otherwise Perl will complain. For instance, if you only overload '==' and then try to use the '+' operator, you get the Operation "+": no method found. Using fallbacks, Perl will try to figure out what you want to do for certain overloads, of which stringification is one of them.

Update: The minimal set of overloaded operations may be helpful.