How did Ty Lee and Yue fall in love?
Like most things, it's an accident of sorts.
In the Boiling Rock, Ty Lee doesn't have much to do, so she stares at the moon through the bars of her cell. She hopes for a lot of things - she hopes that Zuko made it out, that Mai is okay (she's not allowed to talk to or see Mai), that Azula is doing okay without her friends to anchor her, that one day, she'll get out of prison. And she spills her thoughts to the moon, because there's no one else for her to talk to. Of course, she doesn't realize that the moon is listening.
Which means she almost literally jumps out of her skin the first night Yue appears.
There's a pretty short list of things she had expecting, and the literal spirit of the actual moon appeared precisely nowhere on that list.
But Yue listens to her. And accepts her. And treats her like her own person. Like an individual, not just part of a set. It's more than almost anyone else has ever done for her. (And it doesn't help that Yue is absolutely gorgeous.) And they keep talking, night after night, whenever Yue can. And even if she can't be there, somehow, Ty Lee just knows that she's out there, somewhere, listening. And for her, that's enough.
And then all at once, the war ends, and Ty Lee is released, and she joins the Kyoshi Warriors - and even if she's still part of a set, Yue never stops treating her like she's the most important, most special person in the world.
How can Ty Lee can not fall in love with that?
For Yue's part, it starts as simple curiosity. How did this strange girl with an incredible heart end up in a Fire Nation prison? So she listens, and listens, and listens, until finally, she has to know more - that's when she appears in Ty Lee's prison cell, and she nearly jumps out of her own spirit-ly skin when Ty Lee's fist connects with her cheek, because, first of all, that shouldn't be possible, and second of all, the punch was insanely strong.
(It turns out that Ty Lee's spirituality let her be a little more interactive than most with spirits, for better or worse.)
So she starts to ask questions. She starts to come back, every night, and she learns more and more about Ty Lee, the best friend of the Fire Nation's princess, the chi-blocker, the gymnast, the warrior. And the more she learns, the more Yue falls, more and more. She's concerned when Ty Lee joins the Kyoshi Warriors, because they're a matching set, but she watches as Ty Lee grows, and she figures out with Ty Lee that just because they're a matched set doesn't mean they're the same. And finally, Yue gets to see Ty Lee happy.
It does take her a long time until Yue realizes that a huge part of the reason Ty Lee is happy now is because of Yue. Because she started talking to the moon, and the moon talked back, and never gave up on her.
It's a gigantic shock for everyone else - except Suki, because Yue ended up becoming a relatively regular nighttime figure at the Kyoshi Warrior dojo, and she's gotten used to the presence of the new moon spirit - when Yue shows up at night at the Fire Nation palace, arm in arm with Ty Lee. But everyone agrees, seeing Yue again, and Ty Lee being happy, is worth the initial shock of "Yue what in the actual FUCK!?".
Sokka screaming and then fainting when he saw Yue was a bit awkward, though.
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