Clean the seafood:

Using a stiff brush, scrub the clams or mussels well; if using mussels, remove their beards. Place the clams or mussels in a bowl of cool water with a few big pinches of salt and the cornmeal or flour. Place in the refrigerator for at least 3 hours or as long as overnight so that the shellfish will purge themselves of sand. Then drain, scrub again, and rinse well in cool running water.

To clean the squid, holding the body (sac) in one hand and the tentacle end in the other, pull them apart. The two parts will separate easily. Using scissors or a knife, cut off the tentacles just above the eyes; discard everything below the eyes. Use your fingers to push out the small, bony “beak” at the base of the tentacles. Pull out the cellophane-like “spine” from the body and discard it. Peel off the speckled skin. Rinse the squid well in cool running water. Cut the body into ¼-inch-wide rings. Leave the tentacles whole unless they are particularly large; if they are, cut them in half.

Thoroughly pat the clams, mussels, squid, fish fillet strips and peeled shrimp dry. Refrigerate until needed.

Make the sauce:

If using canned tomatoes, drain them, reserving their juice for another use. Using your fingers, push out the excess seeds from each tomato. Chop the tomatoes and set aside. If using fresh tomatoes, slip them into a saucepan of rapidly boiling water and blanch for 30 to 45 seconds. Drain the tomatoes and immediately plunge them into cold water. Drain again and, using a paring knife, lift off the skins and cut out the tough core portions. Cut into quarters lengthwise and, using your fingers, push out the excess seeds. Chop the tomatoes and place in a colander to drain for 5 minutes.

In a large skillet or Dutch oven over medium heat, warm the olive oil. Add the garlic and onion and sauté for 3 to 4 minutes. Add 2 tablespoons of the parsley and continue to sauté until the vegetables soften but do not color, about 7 minutes. Add the tomato paste and wine and continue to cook, stirring, until the alcohol evaporates, about 3 minutes.

Add the tomatoes, oregano, and red pepper flakes and simmer over medium-low heat for 5 minutes. Increase the heat to high and add the squid; sauté for 5 minutes, tossing frequently to ensure even cooking. Add the salt, the clams or mussels, strips of fish fillet, and shrimp and stir to expose all of the seafood to the heat of the pan. Cook until the mollusks open, 8 to 10 minutes.

Stir in the remaining 2 tablespoons parsley and remove from the heat. Do not serve this sauce with grated cheese.