Total Recommendations: 13
While Berserk is much bloodier than Rose of Versailles, both series are about a nation in conflict (the nation in Berserk is in war, the nation in Rose of Versailles is on the verge of war). Both put focus on the backstabbing that happens within the nobility on a regular basis (this plays center stage in Rose of Versailles and takes a very important back seat in Berserk). Neither series shy away from showing the harsh truths of life. Berserk has the "rich people are awful" perspective shown from soldiers, with Rose of Versailles has the "rich people are awful" perspective shown through rich people. If I'm not mistaken, Kentaro Miura cited Rose of Versailles as influence on the political aspect of Berserk. If you're looking for a good drama about crisis, look into these two series.
Both are dark satires whose bleak content is contrasted with the colorful setting. Both also have creatures that can alter the world, but are mostly used for comic relief (fairies in Humanity has Declined, penguins in Mawaru Penguindrum).
Both anime have heavy doses of symbolism, religious imagery and a unique, dream-like atmosphere. The setting in both anime appear to be based off of an old European city, and the location is largely abandoned in both (moreso in Angel's Egg than in Haibane Renmei).
Both anime take place in a strange world (Now and Then, Here and There is presumably Earth near the death of the planet, Angel's Egg is...something else). Both revolve around a mysterious, mostly mute girl carrying an object that may or may not alter the fate of the world. Both also have a staff-carrying boy who tries to convince the mysterious girl to put the object to use. Also, in the few instances where you see technology, expect it to be some sort of military technology (like a tank).
Both are about a plain boy who's new in town who stumbles upon a strange girl who can't take of herself. In both, the girl becomes more emotive and self-sufficient, among other things I won't spoil. The protagonist in Chobits is a bit more perverse than the one in Sakurasou, and Chobits encompasses more genres, however.
Both series revolve an old house filled with oddballs that's in danger of being demolished. There's an outsider (who's more in touch with things) who learns of this and tries to fight for them as the day draws nearer. Both are light fare that's good for a nice night in.
Both series center around people entrusted with an item that can control the fate of the world. Both also have a somewhat blurred line between reality and fiction and it's up to the viewer as to where to draw that line. Also, while neither are comedies, both have some good bits of bizarre humor here and there.
Both series center around a diary that holds the fate of our main characters. Both series also have a psychopath in love (main character in Mirai Nikki, supporting character in Mawaru Penguindrum). Penguindrum is not as straightforward as Mirai Nikki, but both have their twists and turns.
While Now and Then, Here and There is much, much heavier than Trigun, both series feature an idealistic protagonist who believes that all lives are equal and that killing is always wrong. Unfortunately, in both series, their philosophy doesn't do them much good in the cruel desert world in which they live.
At first glance, these two might not be similar, but both are about the endurance of the human spirit and the importance of friendship in harsh times. As well, both feature an inspiring big brother figure whose way of thinking touches the main characters throughout the series, as well as people he hasn't even met. Rainbow is definitely more realistic than Gurren Lagann, but both have a tendency to be uplifting.
Both feature an ordinary person who winds up following people who live on the outskirts of society. The protagonists are only the lesser of two evils in most cases and nobody is really "good." Both Revy and Mugen have a "shoot first, ask questions never" attitude to them. If you like one, you should give the other a try.
Both explore the concept of being there without being there. As well, both series have many things that are not fully explained and would require multiple viewings to understand all of it. While the art style of the two are vastly different (SEL has a more traditional style) and Kaiba has more dialogue, if you liked one of these two, you should try the other.
Both series poke fun at fantasy clichés and feature a delightful cast of oddballs. While Slayers is not as raunchy as Konosuba, chances are if you like one, you'll probably like the other.