Add the tomato, beef broth, water, green onions, soy sauce, rice vinegar, and sugar. Bring to a boil, then reduce heat and simmer for 15 minutes, or until the beef is tender.
Push the soup to the side of the pot and pour in the beaten eggs. Let them cook undisturbed for a minute, then gently scramble them.
Season with salt and pepper to taste.
Serve the soup over cooked noodles.
