Set oven rack about 6 inches from the heat source and preheat the oven's broiler.
Place cream cheese in a microwave-safe bowl; heat in microwave until softened, 1 to 2 minutes. Stir mozzarella cheese and mayonnaise into cream cheese; heat in microwave until mozzarella cheese is melted, about 2 minutes.
Stir basil, green onions, tomato, and garlic into cheese mixture; spread onto sourdough bread slices. Arrange bread slices on a baking sheet.
Broil in the preheated oven until cheese is golden brown and bubbling, 3 to 10 minutes. Cool for 1 to 2 minutes before serving.