In a large saucepan over medium heat, heat the olive oil. When the oil is hot, add the onions and saute until soft, about 5 minutes. Add the garlic and bay leaves and cook for 5 more minutes. Add the tomatoes, 1 tablespoon salt, and thyme and cook for 15 minutes or until the tomatoes soften. Add the water and stock and bring to a boil. Reduce heat to a simmer and continue to cook until the tomatoes are very tender, about 10 minutes. Remove the bay leaves. Using a handheld blender, puree the soup until smooth. Add 1 tablespoon of the lemon juice and hot sauce to taste, as desired. Chill the soup.
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.
This soup can be served either chilled or warm.