From Wednesday, June 21, 2023, at 8:00 a.m. PDT to Sunday, July 2, 2023, at 4:59 p.m. PDT, Chest Form Gimmighoul will be appearing in Pokémon Scarlet and Pokémon Violet Tera Raid Battles. These Pokémon will have various Tera Types, so prepare wisely. If you challenge one in a 5-star Tera Raid Battle, you’ll have a small chance to encounter a Shiny Chest Form Gimmighoul, which can’t normally be encountered in Paldea!
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As of typing, this is currently the only way to legitimately obtain shiny Gimmighoul. Unfortunately, dataminers have discovered that the shiny odds for Gimmighoul are 1/4103.05 which are the shiny odds for normal Tera Raid Battles. Whether this was intentional and if the shiny odds will be boosted in an update to the raid has yet to be discovered, and I will update this answer if any more news drops.
But until then, your only options are to check your 5-star Tera Gimmighoul Raids and reset your raids until you get one, trade with someone who has one, or join an online raid someone is hosting.
As of right now, the only way to get a shiny Gholdengo is by evolving a shiny Gimmighoul or by trading for one.
I shall update this answer if any more news drops on ways to obtain shiny Gimmighoul and Gholdengo.
EDIT: The Gimmighoul Raid Event has ended. Since Gimmighoul is normally shiny-locked in-game, the only way to get a shiny one is by trading with someone who has one!
EDIT 2!
Gimmighoul Returns
Release Dates:
Global: August 9th 2024 - August 22nd 2024
The fifty-forth Tera Raid Battle event offers a challenge to face 1 to 5 Star Tera Raid Battles for Gimmighoul, with 5 Star Raids offering a chance to get Shiny Gimmighoul as a celebration of the Pokémo World Championships
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I'm not sure if the odds are known or not, but I'm assuming they are still the base 1/4103.05 for Tera Raids. I'll update if more news drops.