Heat oven to 375°F. Lightly grease a 9×13-inch baking dish.
Warm olive oil in a skillet over medium heat. Soften onion for 4–5 minutes, stirring.
Stir in garlic, cumin, and oregano. Cook 30 seconds until fragrant, then remove from heat.
Whisk enchilada sauce, sour cream, cream cheese, green chiles, lime juice, salt, and pepper until smooth.
Fold the warm onion mixture into the sauce. Toss half the sauce with the shredded chicken.
Spread a thin layer of remaining sauce in the dish. Add half the tortilla strips, then half the chicken, a layer of sauce, and half the cheese.
Repeat the layers with remaining tortillas, chicken, and sauce. Top with the rest of the cheese.
Cover with foil and bake 20 minutes. Uncover and bake 8–12 minutes more until bubbling and lightly browned.
Rest 10 minutes. Scatter cilantro over the top and serve warm.