In a large pot, heat the olive oil over medium heat. Add the diced onion and sauté until translucent, about 5 minutes. Stir in the minced garlic and cook for an additional 1 minute, being careful not to burn it.
Add the chicken breasts to the pot and pour in the chicken broth. Bring to a boil, then reduce the heat and simmer for about 15-20 minutes, or until the chicken is cooked through.
Remove the chicken from the pot and shred it using two forks. Set aside.
In the same pot, add the green enchilada sauce, black beans, corn, diced tomatoes, cumin, chili powder, salt, and pepper. Stir to combine and let simmer for 10 minutes.
Return the shredded chicken to the pot, mixing it into the soup. Allow the soup to simmer for another 5 minutes, letting all the flavors meld together.
Ladle the soup into bowls and top with shredded cheese. Garnish with fresh cilantro and serve with lime wedges on the side for squeezing over the soup.