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Put the chicken in a large (at least 8-quart) stockpot and add 8 cups water. Bring to a boil over high heat and then reduce the heat to a vigorous simmer. Simmer for 10 minutes, frequently skimming off the foam that floats to the surface.
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Add all the vegetables, the garlic, the cilantro, and the bouillon cubes to the pot, along with the salt and pepper. Stir a few times to distribute the vegetables and submerge as many of the solids as possible. When the broth returns to a gentle boil, partially cover the pot and simmer, stirring once or twice, for 1-1/2 hours. Taste for salt and add more if needed.
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Using tongs or a slotted spoon, pick out the chicken pieces and put them on a large plate. Stir the soup with a large spoon, breaking up some of the potatoes to thicken the soup slightly. Keep hot if serving soon or let cool and refrigerate.
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When the chicken is cool enough to handle, pull the meat off the bones and shred it by hand. Discard the bones and tendons, and put the shredded chicken in a serving bowl.