Process raspberries in a blender or food processor until smooth, stopping to scrape down sides.
Pour puree through a fine wire-mesh strainer into a large container, discarding raspberry seeds.
Bring sugar and 3 cups water to a boil in a saucepan, stirring often.
Remove from heat; add tea bag. Cover and steep 5 minutes.
Remove tea bag with a slotted spoon, squeezing gently; cool tea mixture slightly. Stir together raspberry puree, tea mixture, grape juice, and lemon and lime slices. Cover and chill 2 to 24 hours. Stir in orange soft drink, and serve immediately over ice.