To prepare dashi, combine 8 2/3 cups water and kombu in a saucepan; bring to a boil. Partially cover, reduce heat, and simmer until reduced to about 4 cups (about 1 hour). Strain dashi through a sieve over a bowl; discard the solids.
Place 1/4 cup dashi in a small bowl; stir in miso. Return dashi to pan; bring to a simmer. Add mushrooms, tofu, and ginger to pan. Simmer 10 minutes; discard ginger. Stir in green onions and shiso.