No, unfortunately not. One needs to be at the very least a lawful permanent resident of the US, i.e. a green card holder. TPS is only temporary status. To join the armed forces one needs permanent status in the U.S.
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Yes, if you know one of the languages the army is interested in and the army recruits you.
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Aliens holding TPS status may be eligible to enlist under the Army's Military Accessions Vital to the National Interest (MAVNI) program. Since we are presently under a designated period of hostilities, entering in active duty under this program would automatically make you eligible for naturalization as a U.S. citizen without the need for you to become a lawful permanent resident first. Click on the link below for more information.
While this answer is provided by a Florida Bar Certified Expert in Immigration and Nationality Law, it is for general information purposes only and an attorney/client relationship is neither intended nor created. You should seek out qualified counsel to review your case and provide you with advice specific to your situation. Review Mr. Devore's Avvo Profile for more information about his expertise in immigration law and how to contact him to discuss your case.