Preheat oven to 375°F. Lightly grease a 9×13-inch baking dish. Arrange tortillas on a sheet pan and warm in the oven for 4–5 minutes to dry them slightly (this helps the layers hold up).
Make the filling: Heat olive oil in a large skillet over medium-high heat. Add ground beef and onion; cook, breaking up the meat, until well browned and the onion is tender, 6–8 minutes. Drain excess fat if needed.
Stir in garlic and cook 30 seconds. Add taco seasoning, tomato paste, cumin, and smoked paprika; cook 1 minute to bloom the spices. Pour in the water, then add drained tomatoes with green chiles, black beans, and corn. Simmer on medium, stirring occasionally, until thick and saucy, 6–8 minutes. Season with salt and pepper to taste and stir in the lime juice. Remove from heat.
Warm the refried beans in a saucepan or microwave until loosened; if needed, stir in 1–2 tablespoons water to make them easily spreadable. Mix 1/2 cup Monterey Jack and 1/2 cup cheddar into the warm beans for extra creaminess.
Assemble: Spoon about 1 cup of the beef mixture into the bottom of the prepared dish. Layer 5 tortillas over the sauce, overlapping as needed. Spread half of the cheesy refried beans over the tortillas, top with half of the remaining beef mixture, and sprinkle with about 2/3 cup of the remaining cheeses.
Repeat the layers once more with the remaining tortillas, beans, beef mixture, and all remaining cheese on top.
Cover the dish loosely with foil and bake for 15 minutes. Uncover and bake 8–10 minutes more, until bubbling. For a browned top, broil for 1–2 minutes, watching closely.
Rest 10 minutes before slicing. Scatter cilantro over the top and serve warm.