Bring 8 cups water to a boil. Add the raspberries, sugar, tea bags and basil. Turn off the heat and steep 5 to 10 minutes. Strain and let cool, then combine with the lime juice in a large pitcher. Refrigerate until chilled, about 1 hour. Serve in Raspberry Ice-filled glasses, garnished with fresh basil leaves.
Add 3 raspberries to each ice cube slot in an ice-cube tray. Fill with water and freeze until solid, 6 to 8 hours.