Place frozen raspberries in a saucepan. Add water and sugar then bring to a simmer over medium-high heat then reduce to medium. Simmer, serving occasionally, for 5 minutes until berries are very soft and almost dissolved. Stir in the mint sprigs, then remove pan from heat. Let cool, and discard mint. For a thicker beverage, puree in a blender or using an immersion blender. For a smoother texture, strain and discard solids. Stir in lime juice.
Divide juice among 6 glasses then top with soda water and mint sprigs.