Bring 4 cups water to a boil a medium saucepan; boil for 1 minute. Add 10 of the mint sprigs, the tea bags and the ginger; stir. Turn off the heat and let steep for 20 to 25 minutes. Discard the mint, tea bags and ginger. Stir in the honey until dissolved.
Add 4 cups crushed ice to a pitcher. Pour in the green tea and the lime juice; stir to melt the ice. Add the sake and stir to combine.
Pour the green tea mixture into 4 glasses filled with ice. Garnish with the lime slices and the remaining mint sprigs.