Stir together the sugar and water until sugar is dissolved. Chill until ready to use.
Wash limes well with soap to remove wax and any pesticides. Cut off ends, then quarter. In a large blender, add limes and chilled sugar water. Pulse blender about 6 times. Strain through a fine mesh strainer into a pitcher, pushing down on the pulp to extract as much remaining juice as possible. Discard remaining pulp. Add the coconut milk and sweetened condensed milk to the picture, stirring to combing. Serve immediately with ice.