In a 3 1/2- or 4-quart slow cooker combine hominy, undrained tomatoes, enchilada sauce, onion, garlic, and cumin. Add meat; spoon hominy mixture over meat. Cover and cook on high-heat setting for 2 to 2 1/2 hours.
Remove meat to a cutting board. Stir the 1/2 cup snipped cilantro and lime juice into mixture in slow cooker. Coarsely chop meat; return to cooker and stir to combine.
Serve topped with avocado. Pass tortilla chips. If desired, garnish with additional cilantro, sour cream, and/or lime wedges.