Prep: 5 minutes; Cook: 7 minutes; Stand: 5 minutes.
Bring 2 cups water, 1/2 cup blackberries, sugar, ginger, and lime zest to a boil over medium-high heat in a saucepan, stirring just until sugar is dissolved. Boil for 5 minutes. Remove saucepan from heat, and let stand, covered, for 5 minutes. Pour mixture through a fine strainer set over a bowl; don't press on solids. Reserve the syrup, discarding the blackberries. Then stir in the lime juice.
Add the remaining 2 cups blackberries and blueberries to the syrup. Serve over 1/2 cup lemon sorbet, if desired.