Heat a heavy soup pot over medium heat. Add oil and continue to heat until the oil just begins to smoke. Season the ox tails with salt and pepper, place in the pot and brown on all sides until dark mahogany brown.
Add onion, garlic, jalapeno, tomatoes, apple and water. Bring to a boil, cover, and reduce heat to low; simmer for one hour.
Add chayote and continue to cook for 10 minutes. Add carrots, potatoes, corn and continue to cook for 20 minutes. Add cabbage and cilantro. Cook an additional 10 minutes. Turn off heat and let the soup sit undisturbed for 30 minutes. Skim off any accumulated fat at the top. Serve with warm corn tortillas and a wedge of lime.