Annulments in Royal Marriages
I can think of one marriage in history that was annulled, but the children remained legitimate:
Eleanor of Aquitaine and Louis VII, king of France.
She was only able to bear two living daughters, which meant she was barren (wtf). She convinced Louis and the Pope that the reason she couldn't give Louis sons was because they were cousins (I think to the fifth degree) which was considered cause to set aside a marriage or Plight Troth (I think cousin to the seventh degree was considered cool, but most royals who married were much closer than this. To an extent this may have been intentional since it left a backdoor out of a marriage that didn't turn out as advantageous as originally believed).
Her two daughters kept their legitimacy and Eleanor went on to marry her other cousin to the same degree, Henry II of England and bare him I think four or five sons and a handful of daughters.
Her French daughters were used in marriage as political pieces and the subsequent marriage of their father to another woman, and that marriage producing King Phillip, was not to their detriment.
If Rhaegar could convince a Septon that his wife could not fulfill her purpose (children) I'm sure they would figure out a loophole that could allow him to remarry and set aside his previous wife while protecting his children.
Oh, and Eleanor was the one who really pushed for the annulment, King Louis was originally supposed to be a monk (and was raised to be celibate and pious from an insanely young age), but his brother died in a hunting accident. He was an insecure king and thought women were temptress daughters of Eve. Being married to the most beautiful (richest) woman in the world was very difficult for the both of them. Although he did seem to care about her, I think she was too much for him, and he eventually let her go.
Being a "spiritual relative" such as Godmother or Godfather could also get you out of a marriage, since it was spiritual incest. I didn't hear of that one ever being invoked though.
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