The scenario is not unique. A same-sex couple gets married in a state that allows the practice, such as Minnesota, then moves to one which does not. When they decide to split, the second state is resistant to grant a divorce to a union they don't technically recognize.
The latest example comes from a couple in Kentucky that had previously wed in Massachusetts. The couple has put down roots in Kentucky, where they have jobs and reside. According to them, they should not have to move and leave all this behind just to get a divorce. While states that do not grant same-sex marriage have also denied same-sex divorces, a local legal expert thinks that practice is susceptible to change. According to a law professor in St. Paul, many of the laws simply say that same-sex couples cannot receive the benefits of marriage.
But, since divorce wouldn't usually be considered a benefit of marriage, states could grant divorces to same-sex couple while still refusing to grant the marriage itself.
In Minnesota, this is less of an issue as the state recognizes same-sex marriage. The underlying issue, however, must be understood by local couples. All states, including Minnesota, have a residency requirement to divorce. That is, a person must live in a state for a certain amount of time before they can get a divorce in that state.
If a couple splits and one party moves out of the state, that person can potentially force the other party to go through divorce proceedings in another state. As a result, individuals separated from their spouse may want to contact a local St. Paul attorney to discuss residency issues and divorce strategy.
Source: Daily Journal, "Same-sex couple married in Massachusetts seeking right to divorce in Kentucky," Nov. 4, 2013