Cook and stir bacon in a large pot over medium heat until crisp, about 10 minutes. Drain off grease, leaving about 1 tablespoon in the pot. Break bacon into pieces in the pot.

Place onion in the pot and cook until transparent, about 5 minutes. Mix in potatoes and cook until they begin to turn translucent, about 10 minutes more. Stir in chicken broth and season with oregano, salt, and pepper. Bring to a boil over high heat. Cover and simmer over low heat until potatoes are soft, about 30 minutes.

Pour in cream-style corn and whole kernel corn and cook for 10 more minutes.

Stir in cream, Monterey Jack cheese, and Cheddar cheese over low heat until the soup is hot and the cheeses have melted. Do not boil. Remove from the heat and serve.