Combine 5 cups of the water and the mushrooms in a bowl. Combine the remaining 5 cups water and the wakame in another bowl. Let each stand for 1 hour, or until each is tender.

Using a slotted spoon, transfer the mushrooms to a work surface and thinly slice them. Using the slotted spoon, transfer the wakame to the work surface. Cut away and discard the center veins, then cut the leaves into bite-size squares.

Using a coffee filter or lining a sieve with several layers of cheesecloth, strain the mushroom and wakame soaking liquids into a heavy stockpot. Add the mushrooms, wakame, cabbage, radish, celery, carrots, and onion. Cover and bring to a simmer over high heat. Decrease the heat to medium-low and simmer, stirring occasionally, for 20 minutes, or until the vegetables are very tender. Remove from the heat.

Prepared up to this point, the soup will keep for 1 day, covered and refrigerated. Before continuing, bring the soup to a simmer, stirring occasionally. Just before serving, rest a sieve atop the pot of hot soup, submerging the bottom of the sieve, dissolving it into the soup. Stir the soup to fully blend in the miso. Don't allow the soup to boil vigorously once the miso has been added, since doing so destroys some of the nutrients in the miso. Ladle the soup into individual bowls. Sprinkle with the green onions and serve.