It's intensely funny to me that the two branches of christianity that most people call "catholic" and "orthodox" both claim to be both catholic and orthodox.
Are you sure? Because Eastern Orthodox (or Greek Orthodox, etc) ≠ Byzantine Catholic. The Latin Rite isn't the only version of Catholicism, there's Byzantine Catholics too. Andy Warhol was a Ruthenian/Byzantine Catholic, for example, but since that Church is in full communion with the Roman Catholic Church it's basically only a difference in aesthethics. It's all very confusing