For the brine: Crush the spices with the bottom of a saucepan. Put spices in a saucepan and toast, swirling pan over medium heat for 3 minutes.
Add water, 3 tablespoons salt, brown sugar, ginger, and garlic to the pan. Bring to a boil, remove from heat, and stir in the rum; cool to room temperature.
Remove silver skin from tenderloins and place in a bowl or shallow container.
Pour the brine over the tenderloins. Cover and refrigerate 1 to 4 hours.
Put the coriander, allspice, and cloves in a small skillet and toast, swirling pan, over medium-high heat until fragrant, about 30 seconds. Crush spices with a heavy pan or a mortar and pestle.
Heat butter in a small saucepan over medium-high heat.
Add ginger, garlic, and jalapeno and cook, stirring, until lightly browned and fragrant, about 3 minutes.
Add toasted spices, vinegar, and sugar and stir util dissolved. Stir in pineapple and salt Bring to a boil, reduce heat to low, and simmer until slightly thickened, about 20 minutes. Cool, then stir in scallions, mint, and lime juice.
Prepare an outdoor grill with a medium-high fire for direct and indirect grilling.
Remove the tenderloins from the brine and pat dry.
Brush the grill grate lightly with vegetable oil. Season tenderloins with coarsely ground black pepper and salt to taste. Grill the tenderloins over direct heat util each side is golden brown, about 6 minutes per side. Move the meat over to the cool side of the grill, cover with a disposable aluminum pan, and cook until an instant-read thermometer inserted in the center reads at least 145 F for medium, about 6 to 8 minutes more.
Let rest 5 minutes before slicing. Slice the tenderloins into medallions and serve with the chutney.
