In a medium bowl, mix the sugar with the liqueur, and lime juice. Let the flavors marry for 10 minutes, stirring occasionally, so the sugar dissolves.
Meanwhile, hull the strawberries, and cut them into quarters.
Mix the coconut with the liqueur, and lime mixture. Spoon a layer of coconut, and then a layer of strawberries into 4 cocktail glasses. Repeat with the remaining ingredients.