Remove leftover stuffing, large pieces of loose skin, and big pockets of fat from the carcass, leaving small bits of meat and clinging skin for flavor. Break the carcass into two or three large pieces so it fits in your pot.
Place the carcass in a large stockpot, pour in 4 quarts of cold water to mostly cover the bones, and bring to a boil over medium heat. Reduce to a gentle simmer, skim off any foam, and simmer about 1 hour until the meat falls off the bones.
Lift the carcass out with tongs onto a rimmed baking sheet and cool 10 to 15 minutes. Pull off the remaining meat, chop it into bite-sized pieces, and discard the bones, skin, and gristle.
Set a fine mesh strainer over a clean large pot and carefully pour the broth through it to catch tiny bones and cartilage. Skim off visible fat or use a fat separator, then return the strained broth to the stove.
Peel and dice the potatoes into 1/2 inch pieces, peel and dice the carrots, dice the onion and celery, shred the cabbage, rinse the barley, and crush the canned tomatoes with your hands or a spoon.
Add the chopped turkey to the strained broth and bring to a boil, then reduce to low. Stir in the tomatoes with juices, potatoes, carrots, onion, celery, cabbage, barley, Worcestershire sauce, and all seasonings, stirring so the barley does not clump.
Simmer gently about 1 hour, stirring every 15 minutes, until the potatoes, carrots, and barley are tender. Add a splash of water if the soup becomes too thick, then remove and discard the bay leaf.
Taste and adjust salt and pepper as needed. Ladle the hot soup into bowls and serve right away with your favorite side.