Combine the sugar, blackberries, raspberries and 4 cups water in a large saucepan; bring to a boil, then lower to a simmer. Simmer, mashing the berries and stirring a bit, until the sugar is dissolved. Remove from the heat and add tea bags and baking soda. Let steep for 10 minutes.
Meanwhile, mash the strawberries in a bowl. Divide the strawberries among the cups of an ice cube tray and freeze until completely frozen, about 4 hours.
Remove the tea bags from the hot water. Add 4 cups cold water to a pitcher, then pour in the tea. Place in the fridge to chill.
When ready to serve, add the strawberry ice cubes; the flavor will intensify as the ice cubes melt. Garnish with mint sprigs.