1. Cut tomatoes into chunks, then purée in batches in a food processor. Rub through a fine strainer set over a large saucepan; discard seeds and skins.
2. Stir sugar into tomato purée. Bring to simmering over high heat, then reduce heat to medium and cook, stirring occasionally, until reduced to 4 cups, about 30 minutes.
3. Nest pan in a bowl of ice water and stir occasionally until tomato mixture is cold. Stir in 1/3 cup lime juice. Add more if needed so you taste both sweet tomatoes and lime.
4. Process mixture in an ice cream maker according to manufacturer's directions. Transfer to a bowl and freeze until firm enough to scoop, about 3 hours.