 If you liked
Shinseiki Evangelion
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 ...then you might like
Gilgamesh
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Both are swimming in psychological issues leading towards the ultimate decision for mankind. Both Gilgamesh and and Evangelion have a focus on characters deep emotional distress and there personal development. Both children and parents play a roll in how the story plays out in each. Though one deals with mecha's and the other with psychic powers both of these series share many similarities and I recommend both, to anyone whom enjoys these plot types.
Both take place in a post apocalyptic world. The series once watching are very similar. Both animes have the Fathers playing a big part and both kinda neglected the main characters, mothers are deceased, and the child ultimately has to make the decision for the fate of the world. The Gilgamesh like the Angels have to be fought to save humanity, and they have characters with deep emotional problems.
Gilgamesh is a visible anime based in some NGE aspects. It even makes an explicit reference to NGE in first episode.
In NGE Seele wants to be God, in Gilgamesh is Enkidu and his followers.
Both have a GREAT character development, emotional scenes that touch, the stories are similar and have many expects in common like clones, world catastrophes/post-apocalyptic.
Others elements in common are humanity feelings and psychology, 2 different sides: the one that wants a world revolution/evolution and other that wants the world how it is.
Even the finals have something in common, both are unpredictable and "choking".
Gilgamesh is very slow paced, so you must have
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Also a post-apocalyptic anime with a mythological / metaphysical science fiction framework. Both strong in psychological, inner turmoil and struggle. Many similar themes regarding supernatural-powered enemies, human power factions with particular agendas, scientific research (e.g. subatomic physics, bioengineering) and philosophical considerations.