Mud and Clay may be played by the same person, but have different spouses. Exodus marriages are absolutely not binding outside of Exodus. Mortals may never marry immortals. Both people wishing to wed. This can be as simple as the officiator asking each person if they want to marry the other person, and then declaring them wed, or as long and involved as the elaborate exchange of vows between elves.
Customs differ depending on race.
While not required, generally a note announcing the marriage to at least the immortal staff makes sure the marriage is properly recorded.