Broth: Peel and devein the shrimp and set the meat aside. Put the shells and other ingredients into a medium saucepan and bring to a boil. Reduce the heat and simmer 20 minutes. Strain the broth through a chinois or seive. Much of the flavor is in the shells, so press and pound with a heavy spoon or pestle until all the moisture has been extracted. Discard shell residue and reserve broth. Stew: Drop the sausage pieces into a large, heated saucepan or casserole and sear over medium-high heat. Stir frequently. Cook for about 10 minutes. The sausage will render considerable fat as it cooks. Pour off and discard all but 1 tablespoon of the fat.

Add the ham pieces and continue cooking, stirring often, for an additional 10 minutes.

Add the peppers, celery, onions, and garlic. Scrape the bottom with a wooden spoon or spatula as the mixture cooks to loosen and incorporate the brown particles stuck to the bottom of the pan.

Cook about 15 minutes.

Add tomato puree and tomatoes. Stir to blend and cook together for about 3 minutes.

Pour in the reserved broth and bring to a boil. Add thyme, bay leaf, oregano, red pepper, basil and black pepper. Cook over low heat for 1 hour. Add rice, cover and cook for 15 minutes.

Add reserved shrimp, oysters, and crab and cook 5 minutes more.

Stir in the green onions. Taste for salt.

To Serve: Ladle from the pot or heated tureen into leated large, shallow soup bowls.