Add strawberries and rum to a blender and puree until smooth. Distribute evenly between 2 tall glasses.
Rinse the blender. Put the remaining ingredients in the blender along with 4 ounces (1/2 cup) of cold water. Blend at high speed until smooth.
Very slowly pour the contents of the blender over the strawberry mixture in the glasses. If you like, garnish each glass with a pineapple wedge.