Heat a large heavy-bottomed skillet over medium heat. Pile in pistachios and pine nuts. Cook, stirring frequently, until light brown and incredibly nutty-smelling, 3 to 5 minutes. Transfer to a food processor.
Add garlic to the skillet and cook, turning occasionally, until softened and browned in patches, about 5 minutes. Remove to a plate to cool.
Peel cooled garlic and add to the food processor, along with basil, and process. With the blade still running, drizzle oil in through the chute. Once incorporated, transfer to a large bowl. Stir in salt, black pepper, and red pepper.
Meanwhile, bring a large pot of lightly salted water to a boil. Cook linguine at a boil until tender yet firm to the bite, about 11 minutes. Reserve about 3/4 cup pasta water. Drain remaining water from pasta and give the colander a good shake or two.
Add pasta and tomatoes to pesto. Give it all a good mix and drizzle in enough of the reserved pasta water to get the nutty pesto to coat the pasta. Sprinkle with Parmigiano-Reggiano cheese.