In a large saucepan over medium heat, combine the broth, vinegar, mirin, honey, no-tomato sauce, aminos, ginger, salt, pepper, and garlic and simmer for 5 minutes.
Stir in the mushrooms and bamboo shoots and simmer another 5 minutes.
Meanwhile, stir together the arrowroot and cold water. Add to the soup along with the lime juice, stirring constantly until the soup thickens.
Remove from the heat and continue stirring while you pour in the eggs. Serve garnished with green onions.