 If you liked
Kono Minikuku mo Utsukushii Sekai
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 ...then you might like
Elfen Lied
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When you watch this you will instantly make the connections to Elfen Lied and also the differences as it moves along in the story and shows its own path on a somewhat similar idea.
Both series have a young boy find a mysterious girl with a hidden darker personality (which arouses jealousy from their respective female cousin), and humorous ecchi scenes occaisionally divert attention from the characters' traumatic childhood experiences.
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Recommended by Dorcas_Aurelia
Elfen Lied and This Ugly Yet Beautiful World are a perfect pair. In both we have non-human female characters with a dual personality and a potential for being highly destructive that suddenly find themselves surrounded by humans. With no memory of the past, these girls provide the drama with plenty of nudity along the way. This Ugly Yet Beautiful World had many comic moments that eventually give way to deeper reflections while Elfen Lied is known for its violent content and emotional tragedy.
Elfen Lied and Konomini both share a similar main character, one that is struggling to understand her place in the world all the while trying to go against fate and control the darker side of herself. Besides the bigger picture, relationships also play an important part in both stories. (Jealous cousins included in both.) Lastly, both series can go from idyllic to dark and disturbing in the snap of a finger. (Though Elfen Lied more so.)
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Recommended by blank1234567
Want a anime with a bad girl that has another personality/side of her?
Well then you're gonna like this. Both main characters are very similar because they both are innocent and they both have another side of them. And also both Animes include Nudity so watch out.
Both are similar in some aspects in regards to how an amnesiac girl (or girls) wind up living with some dude, and then there's some crazy shit about the end of the world and whatnot.
Both stories center around the mysterious appearances of a strange yet beautiful girl who crosses paths (and falls in love) with a seemingly-average Japanese high school boy and his family & friends. The main heroines are cute, buxom, and sometimes naïve, harboring sinister powers and darker personalities underneath their innocent appearances. Whereas Elfen Lied is heavy in violence and drama, Kono Minikuku mo Utsukushii Sekai is fluffier and funnier.
Both have a supernatural girl who is being chassed by a strong enemy,and then they meet a guy who the drag into their wolrld of chaos without even knowing why they are being hunted.
The Heroine are the same...All I can think of...
They both have another personality hidden
Both loses their memories
Both have easily noticeable connections when the plot elements follow a similar path: sweet and whimsical girls with hidden dark personalities try, and initially fail, to find acceptance in the cruel, "ugly" world.
To me, Kono Minikuku mo Utsukushii Sekai (or "This Ugly Yet Beautiful World") is pretty much Elfen Lied without the gratuitously violent shock effect.
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Recommended by Kilnerbanker