There is a chance for a Quad-Element Monster to upgrade into a Primordial during Dish-Harmonization if you have collected all other Monsters on an Ethereal Islet and if you have at least 3 of each of the 5 Ethereal Elements in the Dish-Harmonizer.

Each Dish-Harmonization attempt towards collecting a Primordial will fill the meter on the Primordial icon in the Dish Wish menu. This icon changes to reflect the Element of the target Primordial on the Islet you are currently on. Once it reaches 100%, the next time Dish-Harmonization results in a Quad-Element Monster, it will automatically become a Primordial!