tl;dr this is essentially Gainax fanservice. They have close ties to Anime Tenchou and its creator and they were also responsible for both C3bu manga series so this is filled to the brim with throwbacks to Gainax-related things, using the 10th Anime Tenchou anniversary as the reason to make this crossover.
I was quite surprised to learn that the prize they are fighting for is an actual thing, too.
