Place strawberry-pineapple shrub ingredients (water, sugar, strawberries, pineapple, and balsamic) in a medium saucepan. Bring the mixture to a low simmer over medium heat, stirring frequently, until the sugar has dissolved. Remove from the heat and set aside to cool.
Steep tea bags in hot water and cool to room temperature. Once cooled to room temperature, add ice. Strain shrub into the tea (alternatively, you do not have to strain, adding the cooked pineapple and strawberries to your tea) and stir to combine.
Fill serving glasses with ice and pour in the iced tea. Garnish glasses with fresh strawberries and pineapple before serving.
Makes 1 gallon (3.5 L) of iced tea