Pour about 2 inches of water into an 8- to 10-quart, tall, narrow pot and bring to a boil over high heat. Add the potatoes, cover, and cook until barely tender, 10 to 15 minutes.
Meanwhile, in a small bowl, whisk together the ketchup, horseradish, Pickapeppa sauce, Worcestershire sauce, and lime juice.
Add the clams, mussels, shrimp, and corn to the pot, in that order. Cover and steam until the clams and mussels have opened and the shrimp are pink and firm, 5 to 8 minutes more.
With tongs and a large slotted spoon, transfer the corn, seafood, and potatoes to a large bowl. Serve with the cocktail sauce on the side.