"Coincidentally, that's where the tragedy of the commons arises from."
No, the tragedy of the Commons originates in Hardins misunderstanding. Because what he wrote about was the tragedy of the untended commons, or in modern words, no commons at all.
Hardin described a land where there was no rules (or no control of those rules), which is the opposite of what a commons constitutes. He decribes a land that everyone treated as his uncontrolled private property (both no access control and no "dont do bad things" control) and so it failed. Rivalry.
In a commons (cooperation) there are rules, given and enforced by everyone. A commons is always the continued process of managing it together with each other. There is no commons without commoning. Which is especially true to those commons that arent something you can touch like land or the sea (where, currently, there is no commons, just lots of state and private actors who dont have rules that are enforcable or respected by everyone - such the overfishing).
If you are interested in commons, search for books from Silke Helfrich. The 2 Commons books have a big set of diverse articles.
Fair enough, and I'll happily walk back that last statement until I read through that material. I have a fairly basic understanding of the commons, enough to get by, so I can always stand to learn more.
However, that does nothing to invalidate anything I said about private property. Exercising exclusive control over a resource, such as a parcel of land, your body, or virtually anything else, constitutes private ownership. This does not require nor presuppose a state existing in order to occur.