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 as 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, then serve.
Video
Notes
I highly recommend Pomi Strained Tomatoes or Cento tomato puree. While I usually try to stick with local ingredients, these Italian brands are sweeter and milder than American canned tomatoes, which tend to have an acidic aftertaste. If you use an American brand, taste the soup and see if the acidity of the tomatoes needs to be balanced out with a small amount of sweetener.
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 an airtight container in the fridge for up to 4 days. Stir them well before serving. I don't recommend freezing this soup. Its texture won't survive freezing and thawing.