Here is the full list of all 40 Mython Island achievements.
Obtained your first Mython.
Started your Island Adventure.
Found a Hidden Item Capsule.
Visited the Item Shop in your Island Adventure.
Bond formed with a Wild Mython.
Found a Radiant Mython in your Island Adventure.
Read a Info Sign.
Visited the Place of Power and Evolved a Mython in your Island Adventure.
Won a battle against another Island Traveler.
Restored Mython HP and AP at a Healing Shrine in your Island Adventure.
Inflict or Receive Damage that was Super Effective.
Used the Teleporter Device.
Cleared a Run of True Survival Mode.
Successfully Completed Mython Count Activity.
Found All Hiding Roset in Calm Meadow.
Got a Bite while Fishing for Mython.
Cleared the Path in Castle Avia.
Restored Power at Surge Domain.
Complete the Mython Log with every obtainable Mython.
Restored Power to a Shrine in the Old Grounds.
Cleared the Prequel Story of Mython Island.
Cleared Island Adventure Story.
Complete a Survival Run without losing a single Mython.
Took down the Kapslap in Lush Hollow.
Unlocked the Sealed Doors in the Old Grounds.
Defeated the Knight known as Base and received the Basic Shard.
Defeated the Knight known as Gaia and received the Grass Shard.
Defeated the Knight known as Aqua and received the Water Shard.
Defeated the Knight known as Kindle and received the Fire Shard.
Defeated the Knight known as Static and received the Electric Shard.
Defeated the Knight known as Seismic and received the Earth Shard.
Defeated the Knight known as Turbine and received the Air Shard.
Defeated the Knight known as Astra and received the Lunar Shard.
Defeated the Knight known as Rift and received the Void Shard.
Collected 50 Void Shards in Island Adventure Mode.
Collected 100 Void Shards in Island Adventure Mode.
Bonded with Imita in Island Adventure Mode.
Defeated the Primitort guarding the Battle Peak.
Defeated the Craboax in Shore Village.
Defeat the Super Boss Potentia at the Prism Shrine.