1. In a 4- to 5-quart pan over medium-high heat, frequently stir onions, garlic, and oil until onions are limp, about 6 minutes; if necessary to prevent sticking, add 1 to 2 tablespoons water.
2. Add chilies and cumin; stir 30 seconds, then mix in broth, corn, and tomatoes. Cover and bring to a boil over high heat; reduce heat and simmer 5 to 10 minutes to blend flavors.
3. Stack tortillas, cut in half, stack again, and slice crosswise into 1/4-inch-wide strips. Place strips in soup bowls. Ladle soup on top, sprinkle with cheese, and garnish with cilantro.