½cupfirmly packed mint leaves plus extra for garnishing
In a medium sauce pan, boil peaches, sugar, and water for about 3 minutes. Smash peaches while mixture is boiling. Remove from heat and let sit for 10 minutes. Strain peaches out of syrup and into jar. Place in the refrigerator for about one hour to get slightly chilled.
In the bottom of 4 glasses, muddle mint leaves using the back of a spatula to release some of the mint and break leaves up. Add ice to each glass. Set aside.
Place peaches, peach simple syrup, lemon juice, lime juice, and rum in a blender or food processor until pulverized. Pour peach mixture over ice. Top with soda and more mint. Enjoy!