The real solution to this is to make your objects immutable. Don't provide any methods that change the value of your object, and instead return new objects. You could then remove your += and similar overloads and = overload, and just rely on creating new objects from your + overload.
It would actually be possible to still provide the += overload as an optimization if you were providing the = overload. Since the only way to trigger mutation of the object would be via the overload, perl would protect you from ever modifying a shared object by cloning using the = overload. I'm not sure that the extra complexity here would really be worth it.
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