Heat the oil in a large saucepan over medium-high heat.
Add the onion and poblano and cook, stirring frequently, until they begin to soften, 4 to 6 minutes; stir in the cumin.
Add the tomatoes (with their juice), the beans, 2 cups water, 1¼ teaspoons salt, and ¼ teaspoon pepper and bring to a boil.
Reduce heat and simmer, stirring occasionally, until slightly thickened, 12 to 15 minutes.
Add the turkey and cook until heated through, about 3 minutes. Serve with tortilla chips, if desired.