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The film was given an American release at some point, complete with an English dub, predating the adult animated film fad that Ralph Bakshi's Fritz the Cat would later kick-off. However, while the film was a hit in Japan, it flopped in the United States, where it only received a limited release.
To date, the English dub of A Thousand and One Nights has yet to resurface and is believed to be lost. The English dub trailer was on YouTube until recently, when it was taken down for copyright infringement. However, though, on April 30th, 2018, the trailer was made available yet again when it was reuploaded by another YouTube channel by the name of A Lonely Weaboo and is still up to this day.
Update (May 5, 2020): On May 3, 2020, anime distributor Discotek Media announced that they have discovered the English dub. They have restored the dub and will be releasing it on Blu-ray alongside the original Japanese version.