Heat oil in a Dutch oven over medium-high heat. Add onion and ginger; cook, stirring often, until fragrant, about 1 minute. Add cabbage; cook, stirring occasionally, until starting to wilt, 1 to 2 minutes.
Pour in broth, add seaweed and bring to a boil. Reduce heat to medium and simmer for 5 minutes. Add corn, return to a simmer and cook for 2 minutes. Add tofu and cook until hot, about 3 minutes. Stir in miso and cook 1 minute more.
Drizzle eggs onto the surface of the stew and let simmer, undisturbed, until the eggs are just set, 1 to 2 minutes. Remove from the heat and add scallions and vinegar; gently stir to combine.