In a large skillet, heat the olive oil over medium-high heat. Add the onion and jalapeño pepper and cook until softened, about 5 minutes. Add the garlic and cook for 30 seconds. Add the corn and cook, stirring occasionally, until the corn starts to brown, about 8 minutes. Add the tomato paste and cook, stirring, for 1 minute.
Add the canned tomatoes with the liquid from the can, 1 teaspoon salt, the pepper, and the cumin. Cook, stirring occasionally, until the liquid is mostly evaporated, about 15 minutes. Stir in the 1/3 cup cilantro and lime juice.
While the sauce is cooking, make the pasta. Bring a large pot of salted water to a boil. Cook the spaghetti according to the package directions. Save 1 cup of the cooking water and drain the pasta.
Add the pasta to the sauce and toss to combine. If the sauce seems too dry, add some of the reserved cooking water.
Serve the pasta topped with grated cheese and cilantro.