Heat oil in a skillet over medium. Add beef and onion; cook, crumbling, until browned. Stir in garlic for 30 seconds, then drain.
Transfer the beef mixture to the slow cooker. Stir in broth, tomatoes with chiles, black beans, corn, taco seasoning, cumin, and pepper.
Cover and cook on Low for 3 hours, until hot and fragrant.
Stir in cream cheese, processed cheese, cheddar, and evaporated milk. Cover and cook 30 minutes more, then whisk until smooth.
Season with salt and lime juice. If you prefer a looser soup, splash in a bit more broth or milk.
Ladle into bowls and finish with crushed tortilla chips or your favorite nacho toppings.