In a large pot over medium heat, saute the celery and onion in the oil for 5 to 10 minutes, or until onions are tender. Add the chicken broth, beef broth, tomatoes with juice and tomato-vegetable juice cocktail.
Bring to a boil, add the pasta and the barley, reduce heat to low and simmer for 20 minutes, or until the pasta and barley are tender.
Add the mixed vegetables and the corn and bring the soup back to a boil. Then reduce the heat to low and allow the soup to simmer for 15 more minutes.