They could do either of those things but they haven't, and the Governor has said he sees no need to call the legislature into special session to deal with this situation because the Clerks in all but 2 or 3 of the state's counties have had no problem with issuing marriage licenses to same-sex couples.
This is the problem with an elitist oligarchy all 11 of whom attended two Ivy League university’s and are sequestered in a liberal enclave. No different than King George III. At some point enough citizens will have had enough of the royalty and it will end ugly.