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Either put the chicken into a large soup pot whole, or cut it into pieces. Your choice. Add the remaining soup base ingredients plus enough water to cover by 1 inch. Bring this to a simmer and cook gently until the meat is tender, anywhere from 30 minutes for quail to 3 hours for an old rooster or turkey legs.
Remove the bird and strip the meat from the bones. Reserve the meat and discard the bones -- unless you have lots of time, in which case return the bones to the pot and keep cooking for another hour. Whenever you are ready, strain the soup base into a large bowl through a strainer lined with a paper towel. Discard the solids and reserve the broth.
Heat the chicken fat or olive oil in a clean pot over medium heat. Add the onions and stir to coat with the oil. Cook until soft and translucent, but not browned, about 5 minutes. After about 3 minutes of cooking, add the diced poblano or bell pepper.
When the onions are soft, add the garlic and habanero, if using. Let this cook for a minute or two, then add the shredded meat, the tomatoes and the broth. Bring this to a simmer and add salt to taste. Let it all cook for 20 minutes, then add the lime zest and juice. Turn off the heat, and serve the soup with the tortilla chips and cilantro.