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The issue wouldn't be your uncharacterized discharge or the Article 15, directly. However, the underlying facts that led to the uncharacterized discharge may have resulted (and likely did result) in an RE code that wouldn't allow reenlistment.
It is the reenlistment code, often called the RE code, that decides this issue. It would be located on your DD Form 214.
Chances are you are either barred from reenlistment entirely (an RE-4) or you have an RE code that requires a waiver (RE-3 or perhaps RE-2).
Waiverable RE codes are effectively a bar to reenlistment right now also, given the reduced budget and positions for hire. You can't demand or apply for a waiver. Waivers are appeals that recruiters submit, and they only do so when they can't meet their numbers without one.
The only other option would be to try and have the RE code changed, but that is very difficult. You'd have to prove that the basis for your discharge was either an error or an injustice, a very high burden to meet.
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