Place the onion, garlic, salt, chicken, wakame and water in a stock pot. Bring to a boil and simmer until the chicken is cooked through (about an hour). Remove the chicken and cut into small pieces. Return the chicken to the pot and add the tamari, celery, carrots, and zucchini. Bring to a boil again and let simmer until the vegetables are tender.
In a separate bowl, beat the eggs. Slowly whisk them into the boiling soup. Turn off the heat and give the soup a stir. Let cool slightly and serve.