Refrigerate extra ketchup in an airtight container for up to a week.
Preheat oven to 325°.
Combine oil, 1/4 teaspoon salt, and tomatoes in a large bowl; toss gently to coat. Arrange tomatoes, skin side down, on a wire rack set inside a jelly-roll pan. Bake at 325° for 3 hours. Cool slightly; peel. Discard peels.
Combine tomatoes, remaining 1/8 teaspoon salt, basil, and remaining ingredients in a food processor; process until smooth.