Heat the olive oil in a heavy-based saucepan over a medium heat. Sweat the peppers for 7-8 minutes, stirring occasionally, until they start to soften. Add the tuna, chopped basil and adjust the seasoning as necessary.
Meanwhile, bring a large saucepan of salted water to the boil and cook the tagliatelle until al dente; usually 2-3 minutes. Drain the tagliatelle when ready and toss immediately with the tuna and pepper sauce.
Spoon into serving bowls and sprinkle with the grated Parmesan. Garnish with a sprig of bay leaves before serving.