Saudi Arabian women this year will finally get the right to drive. It will just have to be in a post-apocalyptic world filled with baboon kings, crystal giants, fire dancers, mutants and zombie cybersoldiers.

That’s the setting for the coming mobile videogame “Saudi Girls Revolution,” in which a group of young Saudi women race souped-up motorcycles to fight the evil tyrannical rulers of a corrupted Arabian Empire. It is being made by NA3M, a company with offices in Jordan and Denmark whose founder and chief executive is Saudi...