In honor of the Caribbean island of the month, here is a great taste of de islands! This makes much more barbecue sauce than you need for a single chicken. It keeps well in the refrigerator. Adapted from the Cookin' & Grillin'.
Heat the oil in a large saucepan over medium heat. Add the onion, garlic, and chile and sauté until just beginning to brown, about 5 minutes. Stir in the tomatoes and cook until most of the liquid has evaporated, 5 to 7 minutes.
Stir in the tomato paste, barbecue sauce, hot sauce, 1/4 cup each vinegar and sugar, the rum, 1/4 cup water, the oregano, thyme, salt, and pepper and bring to a boil, stirring to dissolve the sugar. Reduce the heat to low and simmer, uncovered, until thick and richly flavored, 20 to 30 minutes, stirring occasionally.
Remove from the heat and taste for seasoning, adding salt, vinegar, or sugar as necessary; the sauce should be highly seasoned. Add water if the sauce seems too thick.
Any unused sauce may be stored, covered, in the refrigerator for 2 to 3 weeks.