Heat oil in a medium saucepan. Cook onion over low heat until translucent, 8 to 10 minutes. Stir in garlic and cook until fragrant. Add blackberries and cook until berries begin to break down, 4 to 5 minutes. Stir in remaining ingredients and 1 cup water. Simmer over low heat for 30 minutes to fully develop the flavors.
Allow barbecue sauce to cool completely. Purée sauce using a blender if a smoother sauce is desired. Refrigerate in an airtight container for up to 1 week.
Per Serving: Serving size: 2 tablespoons, 40 calories (5 from fat), 0.5g total fat, 260mg sodium, 9g carbohydrates, (1 g dietary fiber, 6g sugar), 1g protein.
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