Kombu, dried shiitake mushrooms, miso, and silken tofu are available at Asian and health-food markets.
Combine kombu, mushrooms, and 2 quarts water in a large Dutch oven. Bring to a boil, then reduce heat and simmer 20 minutes. Discard kombu. Remove mushrooms; when cool enough to handle, thinly slice and return to simmering liquid. Whisk in miso.
Add sweet potato to poaching liquid and cook, adjusting heat as necessary to maintain broth temperature at 190 degrees on a thermometer, until fork-tender, about 15 minutes. Season with salt. Remove from heat and stir in tatsoi. Gently add tofu, trying to not break pieces.
Meanwhile, bring 1 1/2 inches water to a simmer in a large, deep skillet. Crack 1 egg into a small bowl. Holding bowl just above surface of water, gently pour egg into skillet. Repeat with remaining eggs. Poach, adjusting heat as necessary to maintain a simmer, just until whites are firm and yolks are still runny, about 3 minutes. With a slotted spoon, transfer eggs to a clean towel. Divide soup and vegetables among bowls; add one egg to each. Season with pepper and serve.