Bring a large pot of salted water to a boil. Add the pasta and cook, stirring occasionally, until tender but still firm to the bite, 8 to 10 minutes. Reserve about 1 cup of the cooking water, then drain the pasta. Place the pasta and 1/2 cup cheese in a large bowl and toss until coated.
In a medium skillet, heat the olive oil over medium-high heat. Add the mushrooms, garlic, 1/2 teaspoon salt and 1/4 teaspoon pepper. Cook, stirring frequently, until the mushrooms begin to brown, 5 to 6 minutes. Add the olives, tomato paste, capers, red pepper flakes, 1 1/2 teaspoons salt, 1/4 teaspoon pepper and the tuna. Using a wooden spoon, break up the tuna into 1-inch pieces. Add the chicken broth and bring the mixture to a boil. Remove the garlic and discard.
Pour the tuna mixture over the pasta. Add the remaining 1/2 cup cheese and the basil, and toss until coated, using the reserved cooking liquid to loosen the sauce. Season with salt and pepper.