Spray the grill with nonstick cooking spray. Preheat the grill to medium-high or prepare a medium-high fire.
To make the glaze, stir together the chutney and the rum in a cup.
To make the rub, combine the cinnamon, allspice and salt in a cup. Spread the rub all over the pork, then lightly spray with nonstick spray. Place the tenderloin on the grill rack and grill, turning, 10 minutes.
Continue to grill the pork, turning and brushing with the glaze, until pork reaches 160 F (70 C). Transfer to a cutting board and let rest for 5 minutes. Cut into 12 slices.