Place the basil, garlic and olive oil in your food processor bowl. Process into a paste.
Add the tomatoes, buttermilk, salt and pepper. Process until smooth and well-incorporated, about 1 minute, stopping if needed to scrape the sides of the bowl with a spatula.
Transfer to a serving bowl, cover and chill 2 hours before serving.
I highly recommend using either Pomi Strained Tomatoes or Cento tomato puree. If you use an American brand instead, taste the soup and see if you'd like to add a teaspoon of honey to counteract their acidity.