Prep and cook time: About 40 minutes, plus at least 6 hours to chill and freeze
In a blender, whirl blackberries until puréed. Pour through a fine strainer into a bowl, pressing pulp with a spatula to extract as much juice as possible; you should have 2 cups purée. Discard solids. Whisk tea and sugar into purée. Cover and chill until cold, at least 3 hours or up to 1 day.
Freeze in an ice cream maker (at least 1 1/2-qt. capacity) according to manufacturer's instructions until machine stops or dasher is hard to turn. Transfer sorbet to an airtight container and freeze until firm, about 3 hours.
Nutritional analysis per 1/2 cup.