> Operator overloading isn't really "right" or "wrong" per se, but it's probably a bad idea for anything that isn't able to fully implement the contract of the operator, including "associativity", "no side effects", whether exceptions can be thrown, etc.

Operators are not special here. What you are saying is basically "don't claim to implement the interface if you didn't implement it".

This exact problem exists widely in, for example, Java. How often do you see a Java object where someone overrode equals() but not hashCode()? I've seen that problem vastly more often than anyone doing something crazy with operators.