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Complete the indicated task to unlock the corresponding costume.

Gilgamesh casual clothes: Unlock Zero Lancer, then complete Arcade mode as Gilgamesh (and defeat Zero Lancer).
Saber Lily: Complete Arcade mode as Saber Alter.

Beat Arcade with Zero Lancer.

Clear Arcade mode 3 times with the same character to unlock that character's EX mission.

Win 16 times to complete Survival mode.

Complete Arcade mode with any character who has Berserker as their final opponent. Alternately, complete Arcade mode five times.