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Background
You'll meet Idiot Doom Spiral near the fishing village in
Martinaise West (#1). After examining the remains of your car (#2), he'll tell you the story of how you wrecked it. If you subsequently give him alcohol, then he'll also tell you the story of how he became homeless. This will allow you to ask him how he keeps losing things, which will lead to him mentioning that he recently lost a jacket. If you offer to track the jacket down for him, then you'll trigger this task.
Walkthrough
Idiot Doom Spiral (#1) will be pretty sure that he lost his jacket somewhere to the north, which won't narrow things down much. If you ask around the fishing village, then Lilienne (#3) will suggest searching through the abandoned fish market since "drunks are inexplicably drawn to markets."
When you go to the fish market, you'll find Idiot Doom Spiral's
Filthy Jacket hanging on a chain link fence (#5). Interacting with it will give you a description of just how filthy it is, which will cause you to lose a point of Morale (so don't interact with it).
When you hand over the jacket to Idiot Doom Spiral, he'll declare it to be "a disgusting pile of hobo rags," and he'll refuse to take it. If you apologize for the jacket's condition, then you'll lose a point of Morale (so don't apologize for it).
To make the jacket more presentable, you'll need to take it to Isobel the Washerwoman (#4). She'll agree to wash it for you, but it'll take her 15 minutes to complete the job. When she's done, she'll give you a
FALN "Faln" Windbreaker.
This time when you hand over the jacket to Idiot Doom Spiral, he won't remember asking you to find it, and he won't be impressed by the FALN product line, either. This will give you two options: you can force Idiot Doom Spiral to take the jacket, or you can keep it for yourself. Either way, you'll earn 10 xp for completing the task.
1 - Idiot Doom Spiral
2 - Wrecked Car
3 - Lilienne
4 - Isobel
5 - Jacket