Why wouldn't Antarctica (or portions of Antarctica) just become colonies of other countries, and inherit the legal systems of their parent nations?
ah, right, the premise of my question might be wrong! do you think colonies is a more likely outcome? why? (will update the question's description)
Not sure if you have already seen these, and I have not read carefully so am not entirely sure where the current views fully stand, which would clearly have implications should the area become habitable.
https://en.wikipedia.org/wiki/Territorial_claims_in_Antarctica
https://coolantarctica.com/Antarctica%20fact%20file/science/government_antarctica.php
https://en.wikipedia.org/wiki/Antarctic_Treaty_System
However, these do seem to suggest that there would be an existing governance in place -- either with the continent divided and most of it subject to the legal jurisdiction of the territorial claimant or the group of countries signing the Antarctic Treaty (apparently 54).
A quick guess would be that the larger treaty related to environment protections and no military would govern a lot of what any future migrants could do there while immigration into specific territories likely government but the country having claimed territorial sovereignty.
Epistemic status: arm chair thinking
I wonder if an Antarctic mass migration would lead to the UN working closer together to develop international laws, which would consequently increase international cooperation.
ETA: if you think Antarctic would be more likely to host colonies of various countries with their respective legal apparatus, I'd be curious to know why