To make a brine, combine 1/2 cup hot water, 2 tablespoons of the salt and the sugar in a cup until the salt and sugar dissolve. Pour into a large ziplock bag and stir in 1 cup cold water, rum, zest and juice of 2 limes, 3 tablespoons of chopped mint, and 3 minced garlic cloves. Add pork, press out air, seal, and refrigerate 3 to 4 hours.
Heat grill to high. Remove pork from brine and discard brine. Pat dry the pork and coat with cooking spray. Combine paprika, 1/2 teaspoon of cumin, and 1/2 teaspoon of black pepper and rub all over pork. Grill pork over high heat until grill-marked all over, 4 to 5 minutes total, turning often. Reduce heat to medium on a gas grill or move pork to a lower-heat part of a charcoal grill. Cover and cook until meat registers 145ºF on an instant-read thermometer, 5 to 6 minutes, turning once. Remove pork to a cutting board and let sit 10 minutes.
Meanwhile, heat oil in small saucepan over medium heat. Add remaining 6 minced garlic cloves and cook 1 minute. Juice remaining 2 limes and carefully pour juice into pan (it may sputter). Add orange juice, oregano, remaining 2 tablespoons of water, 1/2 teaspoon of cumin, 1 teaspoon of salt, and 1/4 teaspoon of pepper. Bring to a boil and boil 1 minute. Remove from heat and stir in 1 1/2 tablespoons of mint.
Slice pork crosswise and spoon sauce over top. Sprinkle with remaining 1 tablespoon of mint and serve with slices of French Baguette.
Tip: For extra flavor on the bread, brush it lightly with olive oil and grill until toasted golden brown.