Once they begin cropping up, though, it means they can be found pretty well anywhere (as returning to old planets at higher levels causes more powerful enemies to spawn), and that means good experience, loot, and the potential for Borderlands 3 Eridium drops at any time.
They can appear at any time, but it takes a while to progress far enough that they become a concern for players. Anointed enemies are the Borderlands 3 equivalent of rare spawns (moreso than Badass enemies, which appear fairly frequently), and they're also challenging fights with strong rewards for completing them.