Heat vegetable oil over medium heat in a large soup pot, and brown the pork and beef, stirring often, about 10 minutes. Mix in the green bell peppers, onions, and celery. Cook until the vegetables have given up their juice; cover and simmer until the vegetables are tender, about 20 minutes. Stir in beef broth, bamboo shoots, water chestnuts, black pepper, and chili powder.
Mix the soy sauce and cornstarch in a bowl until smooth, and pour it into the soup. Stir until the soup thickens, about 1 minute; cover the chili, and reduce heat to a simmer. Simmer until the meat is very tender and the flavors have blended, about 2 hours; about 5 minutes before serving, stir in the mushrooms and bean sprouts.