Well, since I inadvertently helped start this thread, I should say something, shouldn't I?
Unfortunately, I'm not running Traveller, and have no plans to do so anytime soon. Or play, more's the pity.
Still, knowing what I know about how I set things up, I can figure certain attitudes that would prevail.
First with regard to marriage, a lot depends on the exact nature of the Imperium, which I don't really grok yet. However, if designed by me, and given the realities of the universe (slow comms, etc.) it would be an aristocratic confederacy, basically.
And, for any proper confederacy, the primary rules would involve only trade and aggression between member states.
That said, it would probably require limited 'Full Faith and credit' for most things. Marriages, Inheritance, etc., would be according to the planet they were filed under, at least legally. Of course, there would be laws in place at the various sub-sectors and planets that would limit that, but I think most planets would accept full faith and credit for marriage declarations, just as they would with ship registration and the like.
Of course, this doesn't extend to making those with weird marriage customs feel welcome. I'm reminded of Melissa Scott's 'The Paths of Heaven', wherein is was possible to legally join a group marriage. The 'bad guys' of the first book considered it an abomination, but still kept them together, accepting rule of law.
Similarly with homosexuality - or other 'deviant' behaviours. While they might have no choice but to accept a same-sex marriage declaration, that doesn't mean they have to sell things to that couple.
Basically, IMTU, I think the Imperium would do what it could to not only encourage trade, but also to encourage travel, within the limits of its actual ability to enforce change.
Enough rambling for now.
), the individuals in the case, may or may not wish to dispute the ruling of local courts.
