In a medium pot over high heat, add the chicken, potatoes, rice, celery, garlic, 1 teaspoon of salt, and 7 cups of water. Bring to a boil and reduce to medium heat and cook until the chicken is tender, about 30 minutes, stirring occasionally.
Add the frozen vegetables and cook for a couple minutes. If desired, carefully remove some of the fat that float to the top with a spoon while boiling.
Remove from heat. Remove the chicken and celery with tongs. Place the chicken on a cutting board to cool. Discard the celery.
Meanwhile, in a blender, add the cilantro and ½ cup water. Blend until the cilantro is minced. Stir into the soup.
When the chicken is cool enough to handle, shred or cut the chicken into bite size. Discard the bones. Place a desired amount of chicken meat back to the soup. Save the rest for some other use.
Bring the soup to a simmer over medium heat and cook for 5 minutes. Make sure the potatoes are cooked through. Adjust seasoning with salt and pepper. Transfer the soup to serving bowls. Serve immediately with a lime wedge on the side. Squeeze desired amount of lime juice into the soup.