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To be eligible to continue to receive retirement, you have to maintain U.S. citizenship. This is because you have to continue to be eligible for military service or status, even in your retired years. While it is exceedingly rare, military retirees are still subject to activation for various reasons.
So, it is clear that losing your U.S. citizenship would negate your ability to receive retirement pay. Foreign residence is certainly legal and permitted; however, any action taken to relinquish your U.S. citizenship would be considered.
Directly from the DoD regulation concerning this matter, the two things that can be seen as giving up your U.S. Citizenship are:
A. Obtaining citizenship in a foreign state upon his or her application or upon an application filed on his or her behalf by a parent, guardian, or duly authorized agent
B. Taking an oath or making an affirmation or other formal declaration of allegiance to a foreign state or a political subdivision thereof.
Now, that being said, the regulation also recognizes the concept of dual citizenship, noting that a person acquiring foreign citizenship but does NOT relinquish U.S. citizen does not, alone, necessitate losing retirement pay. It's a case by case determination, looking at the actions of the individual, their intent concerning their U.S. citizenship, any military service obligations required for the foreign citizenship and the country involved. For example, joining ISIS or North Korea would probably invalidate the retirement. New Zealand is more neutral and if you maintained U.S. Citizenship, then the pay could continue.
Here is the full DoD regulation:
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