Take one shot glass full of ice cubes and put them inside a shaker filled halfway up with crushed ice. Add 2 ounces of coconut rum. Shake vigorously until all of the ice melts into the liquid.
Pour about half of the mixture from Step 1 over a large bowl containing 4 ounces of fresh squeezed lime juice. Discard the rest of the mixture back onto the ice cube tray.
Using a muddler, gently crush up some of the limes and add those juices directly to the shaker. Stir well to combine everything together. Taste the drink and adjust the sweetness level by adding more sugar syrup, if needed.
Strain the mixture from the shaker over a large plate or martini glass. Garnish with a few sprigs of fresh mint.