I agree on the issue of property rights, and this is a clear case of "tragedy of the commons." The problem with an ocean is you can "privatize" a chunk of the ocean, but unless you can build underwater fences, you have no way of controlling ingress and egress from that spot.
Ownership would be applied to the fish in the sea themselves, not to areas of the sea. A person owning some number of shares of, say, Atlantic cod would say, "I own x percent of the cod in the Atlantic." Of course, you can't label fish like you can cattle on the open range. Fish are a commodity. You'd instead treat fish coming out of the ocean like power coming off the grid. (The comparison I'm making here is to an honestly deregulated power industry like we have here in Pennsylvania, not to a monopoly power system.)
The shareholders would be the "producers" and the fishermen the "consumers". A fisherman contract with a shareholder to buy some of his fish, up to an agreed-upon limit, at an agreed-upon price. It's still up to the fisherman to catch the fish, but it doesn't matter where he catches them; he's just "pulling them off the grid". The shareholder would be legally unable to sell more fish than he holds the rights to, and the fisherman legally unable to catch more fish than he has arranged to purchase.
Of course, some people are going to cheat. (Poaching will be a problem under any conceivable approach to limiting the catch.) The advantage here is that private citizens suddenly have a monetary incentive to put a stop to it; that's real money coming out of their pockets. If the fisheries are considered an unowned resource, poaching is a victimless crime and so is unlikely to be addressed in a meaningful way.
Ecofreaks would be able to protect all the fish they wanted, if they cared to. It's just a matter of putting up enough cash.