Prepare ingredients: peel the green plantains and cut into chunks; peel and cut the yuca (cassava) into chunks; peel and cut the calabaza (Caribbean pumpkin) into chunks; husk the corn and cut into rounds; chop the onion; mince the garlic; chop the cilantro. Ensure the beef stew meat, chicken thighs, and pork shoulder are cut into chunks as listed.
In a small bowl, combine 1 teaspoon salt, 1 teaspoon black pepper, 1 teaspoon oregano, and 1 teaspoon cumin. Rub this seasoning evenly over all the meat pieces.
Heat 2 tablespoons olive oil in a large heavy pot or Dutch oven over medium-high heat. Working in batches so you do not crowd the pan, brown the beef and pork chunks on all sides; transfer browned pieces to a plate.
Brown the chicken chunks in the same pot until lightly browned on all sides; transfer to the plate with the other meats.
Reduce heat to medium. Add the chopped onion and minced garlic to the pot and sauté until the onion is softened and translucent, about 4–5 minutes.
Return all browned meats to the pot. Add 8 cups water or chicken broth. Bring to a boil, then reduce heat to low, partially cover the pot, and simmer until the beef and pork are tender, about 45 minutes to 1½ hours. Skim any foam or excess fat from the surface as needed.
When the meats are tender, add the plantain chunks, yuca chunks, calabaza chunks, and corn rounds to the pot. Simmer, partially covered, until the vegetables are tender and cooked through, about 20–30 minutes.
Stir in 1/2 cup chopped cilantro during the last 3–5 minutes of cooking. Taste the broth and adjust seasonings with the existing salt and pepper if needed.
Serve the sancocho hot, making sure each bowl has a mix of meat, plantain, yuca, calabaza, and corn with broth.