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.
Well, thats creating a false economy and encumbering a market with an added level of "middlemen". Regulators you might say.
The problem is your "producers" are not producers. They do nothing explicitly to grow supply, they simply reject demand by setting false prices to recoup their investment in the shares.
Let actual fish farmers be the producers. No need to muddy the waters. When fish stocks become insufferably low, the real producers will have a field day (bay day?).
Within territorial boundaries it makes sense to allow property rights for real producers. But a simple allocation of shares for an equity that is neither contained, produced (at large), or otherwise controlled will do nothing but create a market in shares, not protect the profitability of the fisherman.
Your approach sounds very much like that described by US_MilitaryRules, wherein they have to apply ahead of time for an allotment of fish. He explains in his post #111.
Perhaps you approach may avoid this?