Can't remember an anime~

[swiftkillz]

Not sure if this information is correct: I thought the basis of one episode was about a girl who had a disease keeping her from going outside, or in direct contact with sunlight (lupus?) and I think she ended up dying in the end. I may be mistaking on this second part as well, but I vaguely remember everyone in the series living in a city covered by a dome...girls name might have been rein (rain/reign), but it's been at least 10+ years since i've seen it. If anyone could help I'd appreciate it!

[Jasanime]

Possibly Ergo Proxy? Girls name is 'Real' or 'Lil' in English...