Combine water, large dried anchovies, dried kelp, and sliced Korean radish in a large pot.
Bring the mixture to a boil over high heat, then reduce heat to medium-low and simmer.
Remove the dried anchovies and dried kelp from the broth using tongs or a slotted spoon. Discard them.
Add the assorted Korean fish cakes to the simmering broth.
Season the soup with soy sauce, minced garlic, salt, and ground black pepper.
Continue to simmer until the fish cakes are tender and the radish is soft.
Taste and adjust seasoning if needed. Garnish with chopped green onions before serving hot.
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