Heat oil in Dutch oven over medium heat. Add onion and garlic;cook, stirring frequently until golden, about 7 minutes. Add the turkey and cook, breaking it up with a wooden spoon, until browned,about 5 minutes.
Stir in the tomatoes, green chiles, tomato paste, chili powder, oregano, and salt; bring to a boil. Reduce the heat and simmer, partially covered, until the chili thickens slightly, about 30 minutes. Stir in hominy and cook until heated through, about 3 minutes.
Meanwhile, combine the red onions, radishes, and cilantro in a small bowl. Divide the chili mixture among 4 bowls; sprinkle each evenly with cheese, then the radish mixture.