Place the tomatoes, cucumber, red bell pepper, red onion, and garlic in a large bowl. Add the basil leaves.
Transfer half the vegetable mixture to a blender. Blend until mostly smooth but still slightly textured.
Pour the blended portion into a second large bowl. Repeat with the remaining vegetables.
Stir in the olive oil, red wine vinegar, salt, black pepper, and cold water. Mix until combined.
Cover the bowl and refrigerate for at least 2 hours to chill thoroughly.
Stir the soup before serving. Ladle into bowls and top each with a few torn basil leaves.