I actually re-read some parts and I think the time when Michiru changed from hating Yukito to enjoying with Yukito was when she was watching Minagi fix his puppet for him, and she said “thank you” to him for making Minagi happy. Thus, I believe this the most:
Now then abooout Michiru…
First thing I put down on my notes here is that there’s a certain BGM, the one played in the very first part of the VN, that always refers to a “motherly” aura. It always plays when Yukito remembers his mother, for one. But it also played when Minagi was talking about her sister (which, at that point, she must have already knew it was the same Michiru). So, more than just Minagi being a sister to her, I feel that she was a mother to her. A lot of her upbringing was done by Minagi and maybe even parts of her personality were caused because of how Minagi interacted with her over the years.
Which segues to my next topic about Michiru. We all know that she is somehow connected to the girl in the sky that Yukito is looking for. Heck, she says it herself, that she found a small portion of that girl’s dream, causing her to materialize. Her original intent in going to the world was because she felt sad for the girl and wanted to show her some happy thoughts. After a while, though, we see that Michiru herself becomes selfish and wishes to stay even longer with Minagi.
The question is, where is that selfishness coming from? To put it more bluntly, where exactly is Michiru’s personality coming from? Let’s face it, she died before she could even be raised to have a personality, yet when Minagi first meets her, she is a functioning human child. She said it herself, people can’t live without memories, so how was she able to live?
My theory is that Michiru’s personality is originally that of the girl in the sky. We see many instances where she talks as if she is that same girl. Especially in the ending, where she requests Yukito to continue looking for that girl. Also when she mentions that “there’s another me” when referring to that girl (actually, much like Misuzu!). However, her personality diverges once she spends time with Minagi. She develops her own memories. And that is what makes Michiru the Michiru that we know.
I guess, in hindsight, this does seem kind of obvious, but I guess it was something worth thinking about. The real, half-sister Michiru, after all, seems a bit more outgoing than the girl-in-the-sky Michiru, when Minagi first speaks to her, at least.
Another thing worth mentioning was how she was avoiding Minagi’s… rather, her own mother. She probably knew about the mother’s situation. Whether it was something Minagi told her, or something she found out about herself, she knew. And she knew about her getting better. Which is why she was so afraid upon meeting her, because she felt that having her mom see a girl with the name “Michiru” will bring up the past and may just cause the same trauma all over again (and had that really cute smile when she realized she wouldn’t). That aside, though… how did she know?
I find it doubtful that she was a stalker. And at the time her mom got better, Minagi was too engrossed with other things. So, the way I see it, she probably got the information from the girl in the sky as well. She must have some sort of connection with her, after all. Since the girl can see some parts of the world (we don’t know which, though), she probably was closely watching the Tohno family’s mother. So she probably somehow transferred that knowledge to Michiru.
So yeah, if anything, Michiru gives us a closer look to that “girl in the sky” and kind of tells us that she has the ability to project herself into a new person separate from herself, yet still keep that connection with her… And this is going off-topic now, but since Misuzu did say in the common route that she feels there’s another her in the sky… Could Misuzu also be one of those projections? I honestly totally forgot most of the details from the anime, so this is just getting me more excited to find out XD