Bring 4 cups of the water to a boil in a medium saucepan and add the tea bags. Remove the pan from the heat, cover it, and allow the tea bags to steep for 5 minutes.

Remove and discard the tea bags. Add the sugar to the saucepan and stir until dissolved. Add the remaining 3 cups water and stir to combine. Transfer the mixture to a pitcher, and add the lemon juice and rum. Chill thoroughly before serving.

Serve over ice, in tall glasses garnished with mint sprigs.