Assuming that certain classes of nouns when appearing as objects of certain kinds of verbs are implicitly owned by the subject, unless strong contextual reasons exist not to interpret it that way, let's consider some shortcuts to blocking the reflexive possession. This of course gets into something I don't like much, viz. disambiguation, but I'll nevertheless consider it.
An obvious way would be to have something that is simply put an indefinite possessive pronoun:
I washed someone's clothes (implicitly: not mine)However, that's kind of obvious and kind of boring. Let's try some other approach: we could possibly get the same kind of implication by reducing the transitivity of the verb, by for instance demoting the object to some other type of argument:
I washed at the clothes ← not mine
I washed the clothes ← mineAlternatively, we could maybe use some kind of voice-like operation on the verb.
the clothes were being washed by me ← not mine
I was washing the clothes ← mineOne possibility could also be that reflexive possession requires the object to follow the verb (or the subject, in an SOV languages) immediately, and inserting anything else between will disrupt the possessivity:
I washed the clothes ← mine
I washed, then, the clothes ← not mineAn interesting kind of implication of this could be if this gets grammaticalized - you could suddenly have a situation where some verbs sometimes have a voice-like thing that simply alters whether the object is reflexively possessed or not (if the inserted thing grammaticalizes to the left), or a case prefix (if the particle grammaticalizes to the right).
If it gets marked on the noun, this could further lead to situations where some nouns also have a generalized form: [nonposs]-brother: any brother, a brother in general, a male sibling; brother: the brother of some relevant NP, usually contextually relevant. If the marking is on the noun, it also, of course, easily can spread to non-object nouns as well.