Patents are to incentivise inventions. If an HIV patient could patent their antibodies, that would limit their usefulness while also giving no incentive to anyone to invent better ones, since you can't invent them - just wait and hope if they appear. So I certainly hope they can't do that.

Rather, it's the researchers who went looking for them who should be granted any patent rights. They're the ones who could work harder to produce better results and can have their performance improved with more rewards like a patent's monopoly.