[Ed. Note: originally posted on gameFAQ’s now-defunct FFX forums by user pmog here (broken link): http://www.gamefaqs.com/boards/2000008-final-fantasy-x-storyline/21477330?page=34]
There isn’t much detailed information about Belgemine in the Ultimanias. She is a summoner from the distant past who wasn’t able to defeat Sin, and her purpose as an unsent is to train young, inexperienced summoners to succeed where she failed. It’s implied from Belgemine’s words that she once trained Braska in much the same way as she’s training Yuna now. She is also mentioned in one of the Creature Create fiend stories in FFX-2 I+LM. An Aquila says that 200 years ago, people feared that with the way things were going, Sin would eventually eradicate all life on Spira. About 150 years ago, Belgemine secretly built a ship and chose many creatures to sail on it to the edge of the world, in order that at least some creatures escape Sin’s wrath and survive. However, it’s unclear as to whether she was alive or unsent at that time. The particular circumstances of Belgemine’s pilgrimage are never clearly explained (eg. exactly how she died, whether or not she managed to obtain her FA), but it’s at least apparent that she doesn’t know the truth about the FA’s futility.
In a passage on Remiem Temple, all it says is that the temple is now Belgemine’s home. There aren’t further details about when/why/how she came to reside there. All unsents have to live somewhere, I guess, and one uninhabited, remote, yet perfectly intact building is as good as another. I’ve managed to find a bit of the script where she dubs herself the “keeper of Remiem”, and I know that she introduces herself differently at the Moonflow/Calm Lands/Remiem if that’s where you meet her for the first time. However, I’ll withhold comment unless I see a copy of the original Japanese script.
Addendum: a little more information on Belgemine. It’s stated that Belgemine’s role is not only to train summoners’ aeons in actual battle, but also to expound on the issue of “what is truly important for a summoner” at every opportunity. This is especially relevant to young summoners, who are easily swayed by Temple policies and the opinions of others. For example, Belgemine reminds Yuna that only summoners can hope to defeat Sin (after machina fail to do so in Operation Mi’ihen), that they aren’t tools of Yevon and to ignore her given traitor status. Belgemine’s detached perspective comes as a result of being an unsent. As such, she’s somewhat disinterested in mundane affairs like politics, focusing only on the essentials. And that, for a summoner, is purely to defeat Sin.
In the FFX Battle Ultimania, there’s a short story about Yuna’s aeon duel with Belgemine. It explains that Belgemine has long been waiting at Remiem for the appearance of a summoner blessed with the power to defeat Sin (I guess that there are many summoners who try but will never quite be capable of defeating Sin). And from their various training sessions, Yuna might just be the summoner Belgemine’s been hoping for. But there isn’t any mention of whether she believes that the calm she expects Yuna to bring will be eternal. In the Creature Create fiend story I mentioned earlier, the Aquila says that Belgemine foretold of a day when Sin would never return. But again, there isn’t any mention of whether she thought it would be the direct result of a particular summoner doing something differently, or because there must eventually come a day when people have atoned for all their sins, thus making the next successful summoner’s calm last forever (the latter being what most summoners (like Braska or Yuna) hope will happen, should they be able to bring the calm).