Halve the berries and place in a saucepan with sugar and water. Simmer gently over moderate heat for 15 minutes, stirring to dissolve sugar. Let cool and add lemon juice. Puree in a blender until smooth, strain through a fine sieve, and refrigerate in an airtight container.
Combine all ingredients in shaker with ice and shake. Strain into a martini glass, and garnish with a slice of fresh lime and strawberry. Serves 1.