Pour the two rums, lime juice and simple syrup into a large pitcher and stir to blend.
In a 12 ounce glass, add 10 mint leaves and about 1/4 cup blackberries. Using a muddler, crush the mint and berries. Pour 4 ounces of the rum mixture into the glass. Add a splash of sparkling water and stir. Fill the glass with ice cubes. Prepare the remaining drinks in the same manner.