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 the soup to a serving bowl, cover, and chill in the fridge for 2 hours before serving.
Video
Notes
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.If the tomatoes you use have no salt added, I recommend using 2 teaspoons of Diamond Crystal kosher salt or 1 teaspoon of any other salt.You can keep the leftovers in the fridge, in an airtight container, for up to 4 days. Make sure to stir them well before serving them. I don't recommend freezing this soup. Its texture won't survive freezing and thawing.