Heat the oil in a soup pot over medium heat. Add the scallions, peppers, onion, and fennel. Cook, stirring occasionally, until the onion is translucent, 6 to 8 minutes.
Add the garlic and cook for another minute.
Add the white wine, bring to a boil, and cook until the volume of wine is reduced by about half, 4 to 6 minutes.
Add the fish broth, tomatoes, tomato puree, and bay leaves. Cover the pot and simmer the mixture slowly for about 45 minutes. Add a small amount of water, if necessary. Cioppino should be more of a broth than a stew.
Season to taste with the salt and pepper. Remove and discard the bay leaves. Add the clams and simmer for about 10 minutes. Discard any clams that do not open.
Separate the claws from the crabs and cut the bodies in half. Add the crab pieces, shrimp, and swordfish to the soup. Simmer until the fish is just cooked through, about 5 minutes.
Add the basil and adjust the seasoning to taste, if necessary. Serve in heated bowls or soup plates.