Heat the oil in a large Dutch oven or pot over medium heat. Add the onion, garlic, both jalapenos, the poblano, cumin, coriander, oregano and allspice. Cook, stirring occasionally, until the vegetables are softened and the spices are toasted, about 7 minutes. Add the chicken, 1 teaspoon salt and a few grinds of pepper. Cook, stirring and breaking up the meat with a wooden spoon, until it starts browning, about 5 minutes.
Add the beans, canned chiles, broth and 1 1/4 cups water; bring to a boil. Add the torn tortillas to the pot; reduce the heat to medium low and simmer, stirring occasionally, until slightly thickened, about 40 minutes.
Remove the halved jalapeno from the chili and stir in all but a few tablespoons of the cilantro; season with 1/2 teaspoon salt and a few grinds of pepper. Serve with assorted toppings, the reserved cilantro, lime wedges and more tortillas.