Simmer the water, anchovies, and kelp together for 15 minutes over medium heat. Pull the kelp out after the first 10 minutes so it doesn't turn slick.
Strain out the anchovies and kelp. You should have about 1.2 L / 5 cups of clear, pale broth.
Fold each fish cake sheet accordion-style and thread it onto a skewer, the way street carts do, so it holds its pleats while it cooks. If using chikuwa or hanpen instead, just slice thickly.
Add the radish to the broth and simmer 8 minutes, until it turns slightly translucent at the edges.
Add the skewered fish cake. Simmer 8 to 10 minutes, until the fish cake is heated through and has taken on the broth's flavor rather than tasting flat.
Stir in the soup soy sauce and garlic. Taste and adjust with salt — it should taste a little saltier than a soup you'd sip on its own, since the fish cake is fairly plain.
Add the scallions and chili, if using. Simmer 2 more minutes, then take it off the heat.