Blend the strawberries then push the resulting puree through a sieve to remove some of the seeds. Tip the sieved puree into the blender again and add the ice, rum and lime juice. Blend again and divide the mixture between 2 Martini glasses.
Thread the lime slices and strawberry halves onto the cocktail sticks and place onto the edge of the glass, serve immediately.
Taste the mixture once blended, if it's a bit bitter for your tastes, add in a teaspoon of sugar and stir through.