Combine water, sugar, and green tea bags in a large pot. Bring to a boil and dunk green tea bags several times in the boiling water. Discard tea bags. Stir in lemonade and orange juice concentrates until dissolved. Remove pot from heat and allow mixture to cool.
Stir vodka and gin into the cooled tea-juice mixture. Pour mixture into a freezer-safe container and freeze until slushy, 3 hours to overnight.
Stir slush to evenly distribute the alcohol. Fill small glasses with about 1/4 cup of slush; top off with lemon-lime soda. Garnish each drink with a maraschino cherry.