Smash the garlic cloves, sprinkle with a generous pinch of salt and, with the flat side of a large knife, mash and smear mixture to a coarse paste.
Stir garlic paste, the ¼ teaspoon salt, Pepsi, ketchup, vinegar, Worcestershire, chili powder, 1 teaspoon black pepper, ½ teaspoon hot sauce, and allspice together in a small saucepan.
Bring to a boil, reduce heat, and simmer until thickened, about 45 minutes.
Allow to cool slightly, stir in the lime juice and season with black pepper and hot sauce, to taste.