Put 3 tablespoons (1/2 ounce) loose green tea in a heat-resistant carafe or teapot. Heat water to 185 degrees F and pour 3/4 cup (10 ounces) over the loose tea. Steep 4 minutes. Strain tea over 3/4 cup (10 ounces) ice in another carafe or other tempered glass container. Set the carafe aside until ice is melted. Cover and refrigerate up to 24 hours.
Squeeze four lime wedges and drop into highball glass, add sugar and muddle. Bruise 12 mint leaves by rolling them between your palms, rub them on the rim of the glass and drop them in. Fill half the glass with crushed ice, add rum, chilled green tea and stir. Top up with crushed ice and a generous splash of sparkling water. Garnish with a sprig of mint.