Tuna & Vegie Diet. Allowed foods: Asparagus, beets, broccoli, Brussels sprouts, cabbage, carrots, cauliflower, celery, cucumber, lettuce, mushrooms, onions, radishes, sauerkraut, squash, string beans, tomato, turnip. Up to 4 cups per meal with diet dressing. You must eat a 3-4 ounce can of water packed tuna with each meal. Have up to 4 meals a day, having as many different vegetables from the list as you want.