In a 5-quart (4.7 l) Dutch oven or pot set over medium heat, combine the water, carrots, and wakame. Bring to a simmer and cook for 5 minutes.
Gently stir in the tofu and mushrooms, then let it return to a simmer, about 3 minutes. Add the cabbage and return to a simmer, about 3 minutes more.
Meanwhile, place the miso in a 2-cup liquid measuring cup. Ladle a bit of the cooking water from the pot, add it to the miso, and stir until smooth. Pour the miso mixture back into the soup and then stir well.
As the soup returns to a simmer, stir in the ginger, soy sauce, and sesame oil. Remove it from the heat and stir in the spinach and scallions. Let the soup rest while the spinach wilts, 1 to 2 minutes. Ladle the soup into bowls and serve with soy sauce, sesame oil, and chili oil on the side for individuals to add to their bowl if they desire.