In a small saucepan, reduce the plums, honey, and rosemary to a thick jam for about 5 minutes. Puree with an immersion blender until smooth, or transfer to a blender bowl. Let cool. One batch makes about 6-8 servings of syrup.
Fill a glass with ice. In a shaker, combine plum syrup, lemon juice, balsamic, and spiced rum. Add ice and shake until cold, then pour into prepared glass. Top with sparkling water. Sprinkle with fresh blueberries and plum slices. Finish with a rosemary sprig.