Heat the olive oil in a large pot over medium heat. Add the onion and cook until it softens, about 5 minutes. Stir in the garlic and cook for 1 minute more.
Add the cumin, chili powder, and cayenne pepper. Stir for 30 seconds until the spices smell fragrant.
Pour in the black beans, diced tomatoes, and vegetable broth. Bring the mixture to a simmer and cook for 15 minutes.
Stir in the spinach and let it wilt for 2 minutes. Taste and season with salt and black pepper.
While the soup simmers, spread the tortilla strips on a baking sheet and bake at 375°F for 8 to 10 minutes until crisp.
Ladle the soup into bowls and top each serving with tortilla strips, lime wedges, and cilantro.