The interesting thing is that exerting physical control of this place might not be that hard given the lack of anyone else trying to do the same. You'd probably just need to build a runway to be able to bring in supplies etc (no idea how hard that would be obviously).

Once you had control then maybe it would be possible to create a libertarian fantasy land of the kind envisioned by the sea-steading folks. Feels like controlling this desert will be easier than building a floating city.