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I keep the shiny pokemon that I run into randomly separate from the ones I chained.
I'm that guy who uses color coordinated teams when I'm not going on simulators, and some of my favorite pokemons alternate color fits differently at times(like my shiny fucking PURPLE Pinsir)
That said, I love my random shinies, but I never use them. I also never name any of them just so they're separate from all of my nicknamed mons.