In a stockpot, warm dashi over medium-low heat. Smash, peel, and trim garlic. Use a spoon to peel ginger, then thinly slice and roughly chop. Thinly slice and roughly chop garlic; then use the flat part of your knife to press down on the garlic and ginger to further break them down. Continue to mince both together. Set aside.
Slice away the fibrous outer layer of the radish and discard; then cut the radish into bite-size pieces. Peel and trim the onion; cut into bite-size pieces. Trim ends of zucchini; cut into bite-size pieces. (The radish, onion, and zucchini pieces should all be the same size to ensure even cooking.) Set the vegetables aside.
Place the squid on a flat surface and pull the head from the body. Feel around the head for the hard beak; the fishmonger may have removed it for you, but if there is one, pull it out and discard. Run a knife down the length of the body to open, then use a paper towel to wipe away the thin membrane that lines the inside of the body. Slice the body and head into bite-size pieces. Trim and discard scallion roots, then slice into 1-inch batons (cut the denser white parts slightly shorter).
Bring dashi to a boil. Then add about 3 tablespoons of the minced garlic and ginger, followed by the salted shrimp, gochujang, and gochugaru. Add chopped radish and onion, about 2 cups each, and bring broth back to a boil. Drain clams and mussels from soaking water; if you see any that are open, tap them to make sure they close. Discard any clams or mussels that remain open, and add the others to the boiling broth. (Note: If clams and mussels are extra large, use fewer than 6, as Chef Kim does in the video.) Finally, add the zucchini, about 2 cups, and bring stew back to a boil.
While stew is boiling, drain liquid from tofu packages; then spoon large chunks of the tofu into the stew, right from the package. Season with a pinch of salt, soy sauce, and sesame oil. Stir and taste. Broth should be on the salty side to balance out the relative blandness of the tofu, so add more salt as necessary. Bring stew back to a boil; then add squid and scallions. Close the lid and continue cooking, 5 more minutes.
When the stew is ready to serve, crack eggs directly into the simmering pot (one per person). Close the lid and let the eggs poach, 2 minutes. Ladle stew into individual bowls, with one egg for every bowl. Serve immediately with steamed white rice on the side.