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Gay Luigi is a "Pokegay" created by Bot for "Super Mario 64 Bloopers: Operation G.A.Y", a plan to turn the whole world gay using clones of Luigi. He appeared as the impostor for Luigi in the blooper.
Although this is not actually stated, it's most likely assumed that he reappears in both episodes of the Super Pokemon 64 bloopers including "Welcome to Pokemans!" and "Gyms and Badges". This is most likely true because it has the same sprite as the Luigi clones in "Operation G.A.Y".
Gay Luigi resemble sprites from canon Mario games such as Mario & Luigi: Superstar Saga.