In a large pitcher, combine the rum, pomegranate and lime juice together. Place 10 mint leaves into each of the glasses. Add 2 tablespoons or 1 ounce of simple syrup to each glass. Use a muddler (or the handle of a wooden spoon or a long-handle ice tea spoon) to crush the mint, releasing its flavor into the syrup—the more muddling, the fuller the flavor.
Fill each glass 3/4 full with ice. Measure 5 ounces of the pomegranate rum mixture into each glass. Stir with a long-handle ice teaspoon to mix everything together. Top off each glass with a little sparkling water or soda. Garnish with a slice of lime.