Juice the lime with the broccoli.
Put the cucumber, garlic, avocado and half of the tomatoes and basil leaves in a food processor or blender. Pour in the juice and process until smooth, adding a little water to loosen the mixture if necessary. Season to taste and chill for at least 30 minutes before serving.
Finely chop the remaining tomatoes and basil leaves. Ladle the soup into bowls and serve sprinkled with the chopped tomatoes and basil.