Puree all but 3 tablespoons cilantro with 2 cups water, the garlic, chile powder and 3/4 teaspoon salt in a blender until smooth. Transfer to a medium saucepan along with the rice and bring to a boil. Reduce the heat to low; cover and cook until the liquid is absorbed, about 18 minutes. Uncover, stir and let cool 5 minutes.
Meanwhile, bring the black bean soup to a simmer in a small saucepan over medium-high heat and cook until the liquid is slightly reduced, about 3 minutes. Stir in the spinach and return to a simmer. Remove from the heat and cover to keep warm.
Toss the corn, tomato, lime juice, the reserved 3 tablespoons cilantro, 1/2 teaspoon salt, and chile powder to taste in a large bowl.
Warm the tortillas in a dry skillet or in the microwave. Divide the rice, bean mixture and cheese among the tortillas; top with some of the corn salsa. Fold up the bottoms of the tortillas, then fold in the sides and roll up. Serve with the remaining corn salsa.