What is enantiotoky, and why doesn't it exist?

Arrhenotoky, in which females arise from fertilized eggs and males from unfertilized eggs, has evolved multiple times in animals, most prominently in the insect order Hymenoptera. An alternative form of haplodiploidy, in which females are haploid and males diploid-here named enantiotoky-is not known to exist. An illustrated thought experiment shows that if it does evolve, it will be very unstable and is expected to disappear very quickly.