Combine blueberries, ketchup, sugar, blackberry jelly, vinegar, corn syrup, tomato paste, margarita mix, red chile peppers, jalapeno pepper, rum, sea salt, paprika powder, blackening seasoning, soy sauce, garlic powder, liquid smoke, black pepper, and lime juice together in a large pot; bring to a boil.
Reduce heat to medium-low and simmer until sauce is smooth, about 30 minutes.
Pour sauce into a blender no more than half full. Cover and hold lid down; pulse a few times before leaving on to blend. Puree in batches until smooth.
