Heat 1 tbsp. vegetable oil in a large pot (at least 5 qts.) over medium heat. Add onions and cook until translucent, 5 to 7 minutes. Stir in two-thirds of the garlic, 1 tbsp. salt, cumin, and chile flakes and cook 2 minutes.
Add broth, tomatoes, and half the lime juice and increase heat to a gentle simmer; cook 20 minutes.
Meanwhile, pour about 1 in. of vegetable oil into a small frying pan set over medium-high heat. When oil is hot but not smoking, add one-third of the tortilla strips and cook until golden brown and crisp, about 2 minutes. With a slotted spoon, transfer strips to a paper towel-lined baking pan. Repeat with remaining tortilla strips in two batches. Sprinkle with 1 tsp. salt. Set aside.
Purée soup in batches in a blender. Return soup to pot and resume simmering. In a small bowl, toss chicken with remaining lime juice, garlic, and 1/2 tsp. salt. Marinate at room temperature for 10 minutes, then add to soup and simmer 5 minutes, until chicken is just cooked through. Stir in cilantro. Serve hot with tortilla strips and your choice of toppings.
Note: Nutritional analysis is per cup.