Place the pepper-garlic mixture in a blender container or food processor bowl. Cover; blend or process until very smooth. Add the 1/2 cup water, the basil, tomato paste, vinegar, sugar, salt, and black pepper. Cover and blend or process with several on-off turns until basil is just chopped and mixture is nearly smooth. Transfer to a 1-quart saucepan. Cook and stir sauce over medium heat until heated through.