Wipe the kombu with a damp cloth to remove any surface salt. Combine the water and kombu in a medium saucepan and heat over medium until small bubbles begin forming around the edges, 8–10 minutes; do not boil.
Just before boiling, remove the kombu. Bring the liquid to a bare simmer, add the katsuobushi, turn off the heat, and let steep until the flakes sink, 5–7 minutes. Strain through a fine-mesh sieve (line with paper towel or cheesecloth for clearer dashi), then return the dashi to the pot.
Set the pot over low heat and add the tofu and dried wakame. Warm gently until the wakame has unfurled and the tofu is heated through, 1–2 minutes; do not boil.
Turn off the heat. Put the miso paste in a ladle, add a little hot dashi to the ladle, and whisk to loosen. Slide the dissolved miso into the pot and stir gently to combine, keeping the soup below a simmer.
Ladle into bowls and top with green onion. Serve immediately while hot.