#tyrion isn't doing well either after she learned about tysha

[goodqueenaly]

What kind of rights - if any - would a divorced noblewoman have in Westeros? In the case where a husband wants to set aside an infertile wife who hasn't done anything wrong personally, would the ex-husband be required to make sure she was provided for or that her honour isn't damaged, or would she just instantly stop being his responsibility? Likewise, is there anything requiring the woman's birth family to take her in again, other than worrying about optics?

A couple things.

Number one, we have no idea of the extent to which divorce - as a separate concept from annulment - exists in Westeros. When commenting on the issue, GRRM merely said that "divorce simply isn’t common”, but did not elaborate. Is divorce a legal remedy for Westerosi married persons, and if so, by what means can married persons seek divorce? (Let’s not even get into the question of whether these rules apply equally to followers of the Faith versus followers of the old gods versus followers of the Drowned God.) Therefore, it is completely impossible to say whether a Westerosi man could even divorce (again, specifically divorce) his wife, much less what would happen to either party in such a scenario.

Number two, even if we just focus on annulment (which definitely does exist in Westeros), we have almost no idea on what grounds Westerosi married persons can seek such a remedy. The only confirmed reason why a Westerosi annulment can be granted is non-consummation, though given the example of Tyrion and Tysha (whose marriage was definitely consummated) it’s very probable that there are other grounds which can secure a legitimate annulment - we just don't know them all. Therefore, we have no idea whether “a husband [who] wants to set aside an infertile wife who hasn't done anything wrong personally” could even do so, much less what would happen to either party even if such an annulment could be granted. (Again, I’m not even going to begin to touch on how the various faiths of Westeros might differ on this point.)

Number three, we have virtually (more on that in a moment) no examples of a successfully obtained Westerosi annulment where the parties were both aristocrats and went on to live (separate) aristocratic lives. (I don’t count the historical example of King Garland II Gardener, given that we have no idea how he legally justified “putting his own wives aside to marry Lord Lymond’s [i.e. Lymond Hightower’s] daughter” nor what happened to these (unfortunately typically nameless) ladies afterward.) Tywin very pointedly, very specifically did not give one iota of a shit what happened to Tysha after he arranged for that marriage to be annulled (and for Tysha herself to be gang-raped). If Aegon V had any conception for what he planned to do for (or with) Jenny of Oldstones after he had the marriage between her and Prince Duncan annulled, we don’t know yet what those plans might have been. Moreover, Viserys I shut down the idea of annulling Prince Daemon’s marriage to Rhea Royce, while Jaehaerys I and Alysanne vocally resisted Rogar Baratheon’s attempt to annul their marriage in 49 AC, so there is not much to glean from these examples on what might have happened to the brides involve. 

Now, we may learn much more about how the Faith and greater Westerosi society treats formerly married persons in TWOW and in Fire and Blood Volume 2. With Lancel having taken the vows of a Warrior’s Son, Amerei is no longer Lady Lannister. Jaime just barely touched upon this point in AFFC - leaving Amerei at Darry “weeping softly at the dissolution of her marriage whilst letting Lyle Crakehall console her” - though his troubled observation of “the hard looks on the faces of her kin as they stood about the yard” may suggest that the Freys will have more to say in the future on how they (and indeed the rest of the world) think of Amerei now that she is no longer Lancel’s bride. Littlefinger, of course, plans on marrying Sansa to Harry Hardyng, and I think will cite the (well-known) fact that Sansa’s marriage to Tyrion was never consummated to have the local religious authorities of the Vale declare Sansa to be as she was prior to the Tyrion marriage - that is, a highborn maiden fit to be wed to Harry. Too, on a historical level, Baelor the Blessed famously had his marriage to Daena dissolved - and while Baelor himself thereafter imprisoned Daena (as well as her sisters) for the rest of his life, we don’t yet know how the rest of Baelor’s court saw Daena after the news of the dissolution.