Here is the full list of all 23 Lost Eidolons achievements.
Complete Chapter 6: Murder of Crows
Complete the sidequest 'A Commander's Gift'
Complete the sidequest 'Monstrous Studies'
Complete Chapter 4: Break of Dawn
Complete Chapter 3: Breakout
Complete Chapter 27: Power of the Lost, and defeat Balastar
Complete Chapter 27: Power of the Lost, and kill Balastar
Finish the gameply on HARD difficulty and Maniac mode
Win all battles without any allies withdrawing
Complete Chapter 12: Into the Breach
Complete Chapter 16: The Bloody Banquet
Complete Chapter 17: Biding Time
Learn Andrea's true feelings
Learn Johanna's true feelings
Learn Klara's true feelings
Learn Marchelle's true feelings
Learn Sera's true feelings
Learn Albrecht's true feelings
Complete the sidequest 'My Brother's Keeper'
Complete the sidequest 'Reaching Within'
Complete the sidequest 'The Winds of Change'
Complete the sidequest 'No Stone Unturned'
Complete the sidequest 'Malice and Greed'