Place the chicken, broth, onion, poblano, garlic, cumin, chili powder, oregano, and salt in a 5-quart or larger slow cooker and stir to combine. Cover and cook on the LOW setting until the chicken is tender, 6 to 8 hours.
Use 2 forks to shred the chicken in the slow cooker. Stir in the hominy and lime juice. Serve with lime juice or wedges, sliced radishes, fresh cilantro, chopped avocado, or tortilla chips if desired.