Place the lime, mint and Thai basil in a cocktail shaker and muddle well. Add Ginger Simple Syrup, club soda, and rum and shake well. Rim a glass with fine sugar and fill with ice. Pour drink over ice. Float coconut milk on top of the drink.
Combine all ingredients in a small saucepan. Bring to a boil and cook until the sugar is dissolved. Remove from the heat and cool. Once cool, strain out the ginger.