Heat the olive oil in a large pot over medium heat. Add the onion and cook, stirring occasionally, until slightly softened, about 5 minutes. Add the garlic, cumin, 1/4 teaspoon salt and a few grinds of pepper; cook until lightly toasted, about 1 minute. Add the tomatoes, black beans, pinto beans and 6 cups water. Bring to a boil, then reduce the heat to medium and stir. Simmer until slightly thickened, about 15 minutes.
Remove 1/2 cup of the bean mixture to a bowl using a slotted spoon; mash with a fork, then return to the pot along with the rice and 1/4 cup cilantro. Cook until the soup thickens, about 5 more minutes. Season with salt and pepper. Warm the tortillas in a dry skillet or wrap in damp paper towels and microwave; wrap in foil to keep warm.
Top each serving of soup with the cheese and the remaining 1/4 cup cilantro. Serve with the warm tortillas.