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El Hazard: Magnificent World - Original OVA series

Partially inspired by the pulp science fiction of Edmund Hamilton and the old Prisoner of Zenda movie this is the tale of a young man who finds himself transported to another world. A world where it turns out he looks very much like a missing princess. A princess who is needed for public appearances.

(The original OVA series is recommended over the later TV and OVA series, The Wanderers: El Hazard, El Hazard 2 and El Hazard: The Alternative World)

Escaflowne ** TV series

High school student Hitomi has always been good at predicting the future with tarot cards. One day at school she collapses while at track practice and has a vision. That evening she finds out the boy she is in love with is going to leave Japan suddenly decides to tell him how she feels before he leaves. But the evening when she does part of her vision becomes real and in the end she is transported to another world. That is the first episode, to find what happens to her I suggest getting the whole series.