Place turkey carcass pieces into a large soup pot and cover with cold water. Stir in onion, celery leaves, and bay leaves, bring to a boil, and reduce heat to medium-low. Simmer for 2 hours. Strain broth and pour back into soup pot. Pick turkey meat from the carcass and chop meat into bite-size pieces.
Heat butter in a skillet; cook and stir carrots and celery until tender, about 8 minutes. Stir carrots, celery, stewed tomatoes with juice, rice, green beans, chicken bouillon cubes, marjoram, thyme, rosemary, ginger, cayenne pepper, salt, and black pepper into broth. Mix in chopped turkey meat.
Bring soup back to a boil, reduce heat to low, and simmer until rice is tender, 30 to 35 minutes. Remove and discard rosemary sprigs; stir corn into soup. Adjust salt, cayenne pepper, and black pepper to taste. Lay a paper towel on top of soup to absorb excess oil, if desired.