Coat the bottom of a large pot with 2 tablespoons of the olive oil and place it over medium heat. Give the oil a minute to heat up, then add the mushrooms. Cook the mushrooms for about 10 minutes, stirring every 2 to 3 minutes, until they're tender and lightly browned. Remove the mushrooms from the pot and transfer them to a plate.
Add the remaining tablespoon of oil to the pot, then add the onion, bell pepper, and Hungarian wax pepper. Cook the vegetables for about 5 minutes, until they start to soften.
Add the tomato paste and cook it while stirring constantly for about 2 minutes, until it darkens a bit, then add the garlic, cinnamon stick, cloves, and oregano. Cook the mixture for about 1 minute longer, stirring constantly, until very fragrant.
Stir in the broth, bay leaves, hominy, and fire-roasted tomatoes. Return the mushrooms to the pot. Raise the heat and bring the mixture to a boil, then reduce the heat and let it simmer for about 20 minutes.
Make the tortilla strips while the soup simmers. Preheat the oven to 400°F. Cut the tortillas into thin strips. Brush or lightly spray both sides of the strips, arrange them on a baking sheet, and sprinkle them lightly with salt. Bake the strips until crispy and lightly browned, about 8 to 10 minutes. Place the baking sheet on a cooling rack when done.
Once the soup has finished simmering, remove the pot from the heat. Remove the bay leaves and cinnamon stick, then stir in the lime juice, orange juice, lime zest, and cilantro. Season the soup with salt and pepper to taste.
Ladle into bowls, top with tortilla strips, and serve.