While the soup is chilling, in a mixing bowl, combine the lobster tails, mango, papaya, pineapple, lime juice, lemon juice, extra-virgin olive oil, oregano, parsley, scallions, chives, remaining 1/4 teaspoon salt and the pepper and toss gently to thoroughly combine.
When ready to serve, ladle the soup into individual bowls. Float some of the lobster claw meat in the center of the soup and divide the remaining lobster garnish among the bowls, about 3 tablespoons per bowl. Garnish each bowl with some of the creme fraiche and a basil sprig.
Cook's Note: This soup can be served either chilled or warm.