In a large stock pot, combine water, chicken broth, chicken soup base, black pepper, carrots, diced potatoes, and onions. Bring to a boil, and then reduce heat. Simmer 15 minutes, or until vegetables are tender, stirring occasionally.
Add celery and red bell pepper, and simmer for 5 minutes.
Gradually add mayonnaise to hot soup, whisking until smooth. Reduce heat to medium low. Gradually stir in processed cheese, sharp Cheddar cheese, and Swiss cheese; continue stirring until cheese melts, about 5 minutes.
Mix in potato flakes. Remove from heat, and let sit for 15 minutes before serving.
