Pat ribs dry with paper towels and place on a board. Season generously with jerk rub, using your fingertips to work it into the ribs. Place ribs bone side down into a sealable container and refrigerate overnight, preferably, or at least 4 hours. (Note: you’re going to bake the ribs in two pans or glass baking dishes, so I cut the rack of ribs into two pieces.)
Preheat oven to 400ºF. Place oven racks in the top and bottom thirds of the oven.
Place ribs bone side down into foil-lined roasting pans and cover the pans with foil, sealing the edges tightly. Place one pan on the top rack and one on the bottom, then bake for 1-1/2 hours, swapping the pans’ positions halfway through.
Meanwhile, make the jerk sauce. Combine all the ingredients in a blender or food processor and process until smooth. Transfer the mixture to a saucepan and bring to a boil over high heat. Reduce the heat to medium and cook until the sauce has slightly thickened and darkened in color, 10 – 15 minutes.
Also meanwhile, make the pineapple-mango salsa. Combine all the ingredients in a bowl. Make at least an hour ahead to let flavors combine. You can make it even sooner and refrigerate, but let it come to room temperature before serving.
Finish the ribs. Remove the foil and baste ribs with the jerk sauce. Cook uncovered, basting every 5 minutes, until ribs are glazed, about 15 minutes.
Serve ribs with pineapple-mango salsa and remaining jerk sauce.