Here is the full list of all 24 Beyond Reality achievements.
Defeat the Three-Headed Wolf
Defeat the Fire Pixie
Defeat 4 Rare Monsters
Defeat 8 Bosses
Defeat the Iceshackle Cyclops
Defeat the Ancient Demon
Make an Offering
Survive an Explosion
Defeat the Lord of the Winds
Defeat the Final Boss on Expert Difficulty
Defeat the Ice Pixie
Defeat the Final Boss Without KOs
Reach Level 10 With All Possible Party Members
Complete the Chronicle
Defeat the Man-Eating Plant
Defeat the Final Boss on Hard Difficulty
Defeat the Desert Snake
Defeat the Final Boss
Defeat the Expert Thief
Obtain the Three Legendary Weapons
Defeat the Unicorn
Reunite the Group
Defeat the Wirewolf
Defeat the Wyvern