Bring water to a boil in a saucepan over high heat. Steep the tea bags until tea is strong, about 5 minutes. Discard tea bags.
Melt the butter in a saucepan over medium heat. Stir in the rice; cook and stir until grains are coated and begin to turn golden, about 2 minutes. Stir in the tea water. Raise heat to medium-high and bring to a boil; reduce heat, cover, and cook on low until most of liquid is absorbed, 15 to 20 minutes.
Remove pot from heat. Stir in milk, sugar, rum, raisins, cranberries, and pineapple. Cover; return to stove and cook over low heat until mixture is thickened and silky, 35 to 45 minutes.
Spoon into serving dishes and dust with cinnamon or cardamom.