Can a Reservist get stationed abroad with their active duty Army spouse?

Yes, there are reserve units in Japan, Korea and Germany. LTC (Join to see) is correct, my friend in Korea had his fiancé , an Army Reservist, do the augmentee thing for a year. She was an E4. You have to reach out and call. You can also get a letter of acceptance and be assigned to that unit I feel they have her MOS.

You both will have to do the leg work there is no two work station office at HRC that does this. Nor an official program. You may have to work in advance the accompanied tour so you can live down range or on post in quarters.

Find out if she can be an individual mobilization augmentee and see if she can train with an Army reserve unit in South Korea. I once met a Navy individual mobilization augmentee who lived in Taiwan. He would go to Guam and do a whole month of active duty and that would count for the whole year. Not sure what range is she can make with any reserve units in South Korea but I know that we have Army Reserve Personnel who go there on temporary Duty. I know a one-star general who goes on temporary Duty in South Korea all the time. Maybe if she can become an individual mobilization augmentee she can be attached to a unit down there and drill once a month or bust out or whole obligation at one time. Let's see what we find out here.

You can command sponsor her and bring her with you, but the Reserve components don't pay to PCS people in a Reserve status. As everyone else pointed out, she can augment or drill overseas while you both live there.