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From Pokémon Emerald onward, if a Pokémon with Synchronize is in the first place in the party (even if fainted), any wild Pokémon encountered has a 50% chance of having the same Nature as the Pokémon with this Ability. Starting in Pokémon Omega Ruby and Alpha Sapphire, if a Pokémon with Synchronize is leading the party, gift Pokémon (except Pokémon obtained via Mystery Gift) in the Undiscovered Egg Group will always have the same Nature.
This does not apply to roaming Pokémon, Pokémon revived from Fossils, or Pokémon in the Entree Forest. In Emerald, it does not apply to stationary Pokémon (such as Regirock or Rayquaza). It also does not apply in the Pokéwalker.