Yes, that’s the most difficult for me. I’ve never seen anything like that tagged.
cooperative=* has been used 62 times already 
So maybe you could just find some values that describe what you mean.
For a sub-key, how about something like cooperative:governance=volunteer_run?
Actually, with taginfo I found the keys governance=* (1495 uses) and governance_type (8 542 uses)
governance_type=* can be found on the wiki at Tag:boundary=protected_area - OpenStreetMap Wiki while simple governance=* seems to be used mostly on protected_areas in the eastern USA, maybe coming from an import.