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Though Wally needs to be rescued in Part 5 to unlock this achievement, being able to do so require some preparation, starting in Part 2 at the earliest. (You might be able to do this from as late as Part 4, though I haven't tried it.)

1. Once you're on LeChuck's ship "LeShip", get to the ship's bow. (The very front of the ship.) There should be a pair of manacles laying there on the deck. They don't have a hotspot, so you generally can't click on it. However, get your monocle lences out. Hovering over the manacles with the lences, a hotspot appears on the manacles. Click on them, and Guybrush will write down the serial number for the locks.

2. When you get back to Meleê Island in Part 4, go to the locksmith and have a key made from the serial number.

3. At the start of Part 5 you'll enter the monkey head and very soon see Wally hanging by his hands from the manacles. Use the key on them to free Wally, and you'll unlock the achievement!