Combine raspberries, lemon juice, and 6 tablespoons sugar in a nonreactive bowl; let sit for 2 hours to draw out the juices (4 hours for frozen berries). Stir occasionally.
In a small saucepan over medium heat, boil the water and remaining 3 tablespoons sugar until sugar is dissolved. Let cool and add the framboise.
Combine all ingredients except the corn syrup in a food processor and puree. Transfer to a bowl and add the syrup, stirring so it doesn't accumulate at the bottom. Chill, then pour into an ice-cream maker and freeze according to manufacturer's instructions.