Make the tortilla chips: In a small pot over medium heat, heat oil. Frying in two batches, add in half the tortilla strips and fry until golden, stirring frequently, about 6 minutes. Transfer fried chips onto a paper towel-lined plate, then immediately sprinkle with salt while hot. Repeat with remaining tortilla strips. Reserve oil.
Make the soup: In a large pot over medium-high heat, heat 2 tablespoons reserved tortilla frying oil. Add jalapeño and poblano and let cook until slightly charred, stirring occasionally, about 8 minutes. Add cumin, coriander, paprika, pepper, and salt and stir until fragrant, about 1 minute.
Reserve ¼ cup of the diced onion for topping. Add remaining onion and garlic to pot and let cook, stirring occasionally, until onions begin to turn golden, about 7 minutes. Add tomato paste and stir until color deepens, about 1 minute. Add chiles and adobo sauce and stir until evenly distributed and fragrant, 1 minute.
Add hominy, corn, beans, tomatoes, and broth. Bring to a simmer, then reduce heat to low. Let cook for at least 20 minutes; for a thicker soup, let cook longer to desired consistency.
Serve soup topped with crispy tortilla strips, avocado, radish, onion, cilantro, and a wedge of lime.