Preheat the oven to 425 degrees F (220 degrees C).
Place oxtails and garlic cloves in a large roasting pan.
Roast in the preheated oven for 30 minutes.
Transfer oxtails to a large soup pot. Peel the roasted garlic and place in a food processor with celery, carrots, and onion. Process until finely chopped; transfer to the soup pot with the oxtails. Add water to cover the bones. Bring to a boil; reduce heat and simmer on low for 2 hours.
Remove oxtails and set aside to cool. Add broth to the soup pot. Add turnips. Simmer until just barely tender, about 10 minutes. Add okra and corn. Simmer until cooked, about 10 minutes more.
Pick the meat from the oxtail bones and chop. Add meat to soup. Season with salt and pepper. Add hot sauce if desired.
