This Seeking Tech guide will show you how to complete the Mirage Vest side quest in Final Fantasy V.
This quest is only accessible when you gain an airship in the Merged World. When you can fly again, make your way to your Yellow Chocobo. Next, you need to ride him down to a patch of forested area located on the south side of the world map.
Afterwards, get off your Yellow Chocobo and then head inside the forest. In the Phantom Village, make your way towards the hidden backdoor of the pub, and then talk to the person in the storage area.
With the quest active, you then need to ride your Yellow Chocobo around the world until you end up back at the Phantom Village again.
After completing your task, talk to the same person to get the Mirage Vest.