You answer:
Basic Training in South Carolina, Advanced Training in Oklahoma, and now stationed in South Korea. The change in training and station is the reason in each case.
However, the important thing is what state you owe tax to. As an active duty soldier/sailor/airman/marine, your state of record is the one you enlisted in, and you will potentially owe tax to that state until you take active steps to change it. This is true even if you are stationed out of the country. I said "potentially" because some military members will not owe tax because their tax will not reach the limit fixed by that state.
This is why you will hear others talk about making a state like Texas your home of record, because Texas has no personal income tax. However, you won't be able to make a no-tax state your home of record unless you actually live there or have some strong connection to it - you just can't throw a dart to pick it.
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