I've been looking out old anime curiosities. I'm currently halfway through each of three six-parters.
10 Tokyo Warriors (1999 OAV) is pretty poor stuff - I'm not sure I'll bother to finish it. There is a supernatural setup not dissimilar to X ; but there is no useful description, so you're wondering what it's all about for quite a while. I gather that the second half is another arc, with a timeskip and unexplained differences. The arcs are taken selectively from a series of light novels.
Hanbun no Tsuki ga Noboru Sora, known in English as Looking Up at the Half-Moon, also started as a series of light novels. It was adapted as a 6-episode anime TV series, followed the same year by a 13-episode live-action TV series, and a few years later by a live-action film. It's unusual in being set entirely in a hospital (except that the protagonists often slip out for various reasons). It's a romance, with one party having a probably terminal illness, for which they will have a potentially fatal operation - this (but nothing else) is like Your Lie in April. I think the 2006 anime version is rather sweet - quite gentle, and mostly not too absurd. The plot uses Night on the Galactic Railroad, both as a book which is read by the characters, and as having plot elements that could relate to their story; it's more deeply embedded, actually, than the similar usage in the film Giovanni's Island. Reviews vary strongly from loving it to hating it.
Please Save My Earth (original title is in English) is a 1993 OVA with six 30-minute episodes. The story covers less than half of a 21-volume manga. The story involves a group of children who discover that they have the memories of seven now-dead aliens (not necessarily of the same gender) who had been isolated on the Earth's moon when their civilisation was destroyed by an unimaginable weapon. So the story has two parallel paths, tracing the behaviour of the trapped aliens and the realisation of what happened in their pasts by the children. There may be some low-level time-travel shenanigans as well, as one of the aliens is seen to be aware of her reincarnation in Japan. Although the initial build-up is slow, this series is clearly posing moral dilemmas for the viewer to ponder. I'm pretty sure I'll go on to read the manga as well. Music in the anime is by Yoko Kanno, which is an additional recommendation.