In a large Dutch oven or heavy saucepan, heat the 3 tablespoons oil over medium-high heat. Add the onion, garlic, poblano chile, salt, and black pepper. Reduce the heat to medium and sauté for about 2 minutes. Then add the lime juice, cumin, paprika, coriander, bay leaves, red pepper flakes, and cilantro. Continue to cook for 2 to 3 minutes, until the seasonings release their aromas and the onion and chile have softened.
Add the tomatoes and chicken stock and bring to a boil over high heat. Then reduce the heat to medium, partially cover, and cook, stirring intermittently, for about 1 1/2 hours.
While the soup is cooking, heat the remaining 1 cup oil in a small saucepan over high heat. Fry the tortilla strips in batches the hot oil for about 3 minutes, until crisp. Use a slotted spoon or tongs to remove them with a slotted spoon and drain on paper towels.
Once the liquid has thickened somewhat, remove the bay leaves and purée the soup with an immersion blender. (If you do not have an immersion blender, you may carefully purée the soup in batches in a blender, but let the soup cool for at least 10 minutes first.) The consistency may be somewhat thinner than you expect, but the taste should be perfect. (If you desire a thicker consistency, gently simmer the soup over medium-high heat for a few minutes until it thickens slightly.)
Serve the tortilla soup in bowls and pass the tortilla strips, avocado, onion, cheese, lime wedges, cilantro, and radishes on the side as toppings.