Combine tomatoes, mushrooms, soy sauce, and honey with 1 qt. water in a medium saucepan and simmer 5 minutes. Turn off heat and whisk in miso, tea leaves, and salt; let steep 5 minutes. Strain through a fine-mesh colander (save shiitakes for another use, such as fried rice, pasta, or in the dashi for soup). Let cool. Dashi keeps, chilled, in a large glass jar for up to 1 week.