In a large pitcher or bowl, combine the light rum, dark rum, pineapple juice, orange juice, grenadine, and fresh lime juice. Stir well to mix all the ingredients thoroughly.
Chill the mixture in the refrigerator for at least 1 hour to allow the flavors to meld.
Just before serving, add the club soda to the punch for a refreshing fizz. Stir gently to combine.
Fill glasses with ice and pour the rum punch over the ice.
Garnish each glass with slices of oranges, limes, and pineapple, and add a sprig of fresh mint for an extra touch.