Mince or finely grate 2 garlic cloves and transfer to a blender. Add basil, mayonnaise, 3 tablespoons oil, anchovies, and lemon juice. Purée until smooth. Transfer basil rouille to a small bowl, cover, and chill.
Heat remaining 2 tablespoons oil in a large heavy pot over medium-high heat. Add tomatoes and fennel; season with salt and pepper and cook, stirring occasionally, until tomatoes burst, about 10 minutes.
Slice remaining 2 garlic cloves and add to pot. Cook, stirring often, until garlic becomes fragrant, about 1 minute. Pour in wine and cook, stirring often, until almost absorbed, about 1 minute. Add clam juice and 4 cups water and bring to a boil. Add shellfish and cook, covered, until opened (discard any that do not open), about 3 minutes. Stir in parsley. Spread basil rouille on bread and serve alongside.
