Heat the oil over medium-high heat in a large (at least 8-quart), heavy-duty pot or Dutch oven. Add the onion, jalapeño, garlic, and 1/2 Tbs. salt. Cook, stirring occasionally, until the vegetables are softened and lightly browned in places, 4 to 6 minutes. Add the chili powder and cumin and cook, stirring, until they become fragrant, 30 seconds.
Add the broth and bring to a boil. Reduce to a simmer and cook, stirring, until the vegetables are completely tender, 20 to 30 minutes.
Stir in the reserved shredded chicken, hominy, corn, and tomatoes. Simmer, stirring occasionally, until all are tender and the flavors meld, 5 to 10 minutes.
Stir in the cilantro and 1 Tbs. of the lime juice. Taste, and add more lime juice, 1 Tbs. at a time, to taste, up to 4 Tbs. Season with salt and pepper to taste and garnish each bowl with a small handful of crisped tortilla strips, if you like.