Operation: Monkey! (Zanarkand Ruins)
“Operation: Monkey!” is an optional side quest that you can complete during two visits to the Zanarkand Ruins. It never appears in your official mission list, so it’s an easy quest to miss. Start by speaking to Isaaru in the hallway inside the Dome.
Every monkey in the Dome has a soulmate somewhere else inside it. The pairs are matched by their names; for example, Birch goes with Sequoia, Spring with Autumn, Dusk with Dawne, etc. Look for a monkey that has hearts over its head, grab it, then carry it to its partner and interact with them. The two stay monkeys will sit together at the second monkey’s spot for the rest of the quest.
The monkey couples have to be matched in order. If you carry a monkey to a partner further down the list before the earlier couples are done, nothing happens. Match all twelve and you are rewarded with the 
Soul of Thamasa accessory, which is worth +15 Magic.
You can complete the quest in either Chapter 2 or Chapter 3, and the walkthrough covers both visits:
Walkthrough
Speak to Isaaru in the Dome hallway to start the quest, then work through the couples before you leave.
Walkthrough
Your second chance at the quest, alongside the return visit to Isaaru for Story Completion Percent.
The maps below cover the five areas of the Dome and identify each of the monkey couples in the order that you need to match them up. Each number appears twice, once in blue and once in red:
Corridor
10
11
7
9
12
3
Cloister of Trials
6
4
2
5
8
10
Chamber of the Fayth
1
8
7
11
Great Hall
12
9
4
6
2
The Beyond
5
3
1