Does the ability "Synchronize" have any special effects outside of battle?

Question

I thought I read somewhere that it can help get Pokémon with similar natures when breeding, but I can't remember...

Answer 1

If the first Pokémon in your party has Synchronize, you have a 50% chance of encountering a wild Pokémon with the same nature as your Sumchronizer. For example, if I have an Adamant Synchronize Abra at the front of my party, all wild Pokémon have a 50% chance of having an Adamant Nature.

This does not apply to roaming Pokémon, Pokémon revived from Fossils, or Pokémon in the Entree Forest.

Source