Clean and crack the crab, or have your fishmonger do it. Cut the claws and bodies into several pieces.
In a saucepan, heat the olive oil over medium-low heat. Add the garlic and cook until soft, but not golden, about 1 minute. Remove the garlic and discard. Add the wine, increase the heat to high, and simmer until it has reduced by three-quarters, 5-7 minutes. Using a food mill, puree the tomatoes directly into the pan. Add the clam juice, vinegar, torn basil pieces, oregano, red pepper flakes, and salt to taste. As soon as the mixture comes to a boil, decrease the heat to medium-low, and simmer until slightly thickened, 8 to 10 minutes. Add the crab and continue to cook for 5 minutes. Taste and season with salt.
Bring a large pot of salted water to a boil over high heat. Add the pasta and cook until al dente, 5 to 8 minutes. Drain and toss with the crab and tomato sauce. Serve immediately, garnished with the strips of basil.