Place the chile, tomatoes, and onion on a hot grill or directly over a flame and cook, turning as needed, for 3 to 4 minutes, until the skins of the chile and tomatoes blacken slightly and blister and the onion softens in its own smoke. Place the chile in a plastic bag to steam. Peel, seed, and chop the tomatoes. Peel, stem, and seed the chile. Cut the chile into strips.
Heat the oil in a large heavy soup pot over medium heat. Season the chicken and pork with salt and pepper. Add the chicken and pork to the pot and cook, turning once, for about 10 minutes, until browned. Add the chorizo, ham hock, tomato purée, garlic, carrots, mushrooms, and corn and enough water to cover. Bring to a boil and then decrease the heat to low and simmer gently. Skim the surface with a slotted spoon to remove any impurities that accumulate. Cover and cook for about 40 minutes, until the meat is tender when pierced with a fork. Add the grilled vegetables and simmer for about 12 minutes, until warmed through.
To serve, transfer the caldo to a casserole or serving bowl. Squeeze the limes over the stew and sprinkle with the cilantro. Serve immediately.
Beef Caldo with Tripe: Omit the chicken and pork and substitute 4 beef short ribs and 1 pound of prepared tripe, cut into 1/2-inch pieces.