It'd end up being treated in much the same way as the Antarctic I'd imagine.

The Antarctic Treaty of 1959 could very well apply to colonisation of the moon.

Swap Antarctica for the moon as I've done here, and it's all relevant, and noble in its intent:

Article I
[The moon for peaceful purposes only]

1. The moon shall be used for peaceful purposes only. There shall be prohibited, inter alia, any measures of a military nature, such as the establishment of military bases and fortifications, the carrying out of military maneuvers, as well as the testing of any type of weapons.

2. The present Treaty shall not prevent the use of military personnel or equipment for scientific research or for any other peaceful purposes.


Article II
[freedom of scientific investigation to continue]

Freedom of scientific investigation on the moon and cooperation toward that end, as applied during the International Geophysical Year, shall continue, subject to the provisions of the present Treaty.


Article III
[plans and results to be exchanged]

1. In order to promote international cooperation in scientific investigation on the moon, as provided for in Article II of the present Treaty, the Contracting Parties agree that, to the greatest extent feasible and practicable:

(a) information regarding plans for scientific programs on the moon shall be exchanged to permit maximum economy and efficiency of operations;

(b) scientific personnel shall be exchanged on the moon between expeditions and stations;

(c) scientific observations and results from the moon shall be exchanged and made freely available.

2. In implementing this Article, every encouragement shall be given to the establishment of cooperative working relations with those Specialized Agencies of the United Nations and other international organizations having a scientific or technical interest on the moon.


Article IV
[territorial claims]

1. Nothing contained in the present Treaty shall be interpreted as:

(a) a renunciation by any Contracting Party of previously asserted rights of or claims to territorial sovereignty on the moon;

(b) a renunciation or diminution by any Contracting Party of any basis of claim to territorial sovereignty on the moon which it may have whether as a result of its activities or those of its nationals on the moon, or otherwise;

(c) prejudicing the position of any Contracting Party as regards its recognition or nonrecognition of any other State's right of or claim or basis of claim to territorial sovereignty on the moon.

2. No acts or activities taking place while the present Treaty is in force shall constitute a basis for asserting, supporting or denying a claim to territorial sovereignty on the moon. No new claim, or enlargement of an existing claim, to territorial sovereignty shall be asserted while the present Treaty is in force.

~

And there are ten further articles in the Antarctic treaty of varying relevance.