Tenkuu no Escaflowne, Sora wo Miageru Shoujo no Hitomi ni Utsuru Sekai Recommendations
Both anime have a world that is on the brink of destruction and a normal girl from earth meets a prince/king from that other world.
Two worlds that somehow exist in parallel universes one of which is somewhere in the sky (or even space), a plain girl who has some powers to influence the fate of the world.
Surely, Escaflowne is far more better in all aspects but Munto is the closest to it in the idea itself and plot.
girl seeing another parallel world because of her dormant powers. She helps the King from that parallel world by lending her powers. The story is very similar to the Vision of Escaflowne.
Both animes involve a female from earth having a unique power that is needed to help a king in another dimension (mirror of earth) - in order to save it. Escaflowne has more romance involved in the plot and is longer in length then Munto. If you like one, you will like the other.