We don't particularly care who has these items, but we want to remove incentives for randomly digging them up. The archeological value of documenting the context of an item is larger than any analysis of item itself, and once an item is removed to be sold, that contextual value is permanently destroyed.

Gaining an ability to track and possibly recover existing artifacts is nice to have, but it's actually harmful if it means creating a financial motivation for removing new artifacts and thus destroying the context for them; ideally, we want to create a situation where they get left in the ground because it's not worth for the looters to spend effort digging them up.

Maybe we should create incentives for people to sell items with their context.

Then most of them will have to admit the object was looted.The items that are legally acquired are always losted with their provenance.

I don't think there is only one good solution for this. This is a very complex problem, which, in my opinion, needs diverse approaches and combined co-operations of the governments, museums, and international and intergovernmental institutions (like ICOM, for example) and agencies (like UNESCO) and organizations (like UNIDROIT).

There are already a couple of conventions (some of them I have posted in the comment below), which proved to be ineffective in the face of Iraqi (destroyed sites of Nineveh, looted Mosul museum and Mari temple, etc.) and Syrian(destroyed Palmyra) wars and not only (forever lost Buddhas of Bamiyan in Afghanistan). Those international conventions will never be 100% effective, unless all parties are involved and working on the problem and websites like Ebay and the auction houses have to be part of it.

What would be of a great help in this whole story, is if the cyber security specialists (please correct me if I used the wrong term for it)would co-operate and help track down the black markets.

I'm not a specialist myself and I believe there are far more experienced people working on this problem already. I just think that a combination of 'powers' would bring far more effective results.