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1. Heat 1 tbsp of the oil in a large saucepan over medium heat. Add the onion and cook, stirring, for 5 minutes, until softened. Add the garlic and stir for 30 seconds. Transfer to a blender, add the tomatoes, and process until smooth.

2. Return the purée to the pan and simmer for 8–10 minutes, stirring constantly, until reduced by one-third. Stir in the stock and bring to a boil. Reduce the heat, partially cover the pan, and simmer gently for 15 minutes.

3. Place a nonstick frying pan over medium heat. Add the chiles, and press them flat with a spatula until they begin to blister, then turn them and repeat for the other side. Transfer to a plate and cool. Cut into small pieces and set aside.

4. Heat the remaining 4 tbsp oil in a frying pan until hot but not smoking. Add the tortilla strips in batches and fry just until crisp. Remove with a slotted spoon, and drain on paper towels.

5. When ready to serve, add the chiles to the soup and bring to a boil. Simmer for 3 minutes, or until the chiles are soft. Stir in the cilantro, lime juice, and salt and pepper to taste. Ladle the soup into bowls, top with the tortillas and cheese, and serve with the limes.