Heat oil in a large Dutch oven over medium heat. Add onions and poblano chiles; cook and stir until soft, about 5 minutes. Add garlic; cook and stir until fragrant, 1 to 2 minutes. Stir in chile powder, cumin, and oregano.
Add turkey broth, turkey, cannellini beans, hominy, chile peppers, salt, and pepper. Add water just to cover, stir to combine, and bring to a boil. Reduce the heat to low, cover, and simmer, stirring occasionally, until flavors have blended, about 1 hour.
