Use the ripest, juiciest blackberries or raspberries you can find for this cocktail.
fresh lime juice
sprigs Thai basil or sweet basil
Purée blackberries and sugar in a blender. Let stand, stirring occasionally, until sugar is dissolved, about 10 minutes. Strain purée through a fine-mesh sieve into a pitcher (there will be about 3/4 cup); discard seeds in sieve. Stir gin and lime juice into pitcher.
Divide drink among ice-filled glasses; top with soda. Garnish each with a basil sprig.