Cut kernels from corn to yield about 2 ½ cups; set aside. Scrape corn milk from cobs into a large heatproof bowl using the back of a knife; discard cobs. Add egg yolks, eggs, Pecorino Romano cheese, Parmigiano-Reggiano cheese, and black pepper to bowl with corn milk; whisk until well combined. Set aside.

Bring a large pot of lightly salted water to a boil over high. Meanwhile, cook bacon in a medium skillet over medium, stirring often, until browned and lightly crisp, 6 to 8 minutes. Transfer bacon to a paper towel–lined plate using a slotted spoon. Add corn kernels to drippings in skillet; cook, stirring often, until tender, about 2 minutes. Remove from heat. Transfer about ¼ cup corn to plate with bacon. Transfer remaining corn and drippings in skillet to egg mixture; whisk immediately to release steam until slightly cooled, about 1 minute. Set aside.

Cook spaghetti in boiling water according to package directions until al dente. About 1 minute before pasta is finished cooking, scoop out ¼ cup cooking liquid, and slowly pour into egg mixture, whisking constantly, until well combined and sauce is loosened.

Transfer about one-third of pasta to egg mixture using tongs, letting excess water drip off; quickly and gently toss to combine. Transfer remaining pasta to egg mixture; continue stirring and tossing using tongs until sauce thickens and coats pasta, 2 to 3 minutes. (Sauce will continue to thicken as it cools.)

Divide pasta evenly among 4 shallow bowls, and top with reserved bacon and corn. Garnish with basil and additional Pecorino Romano and pepper; serve immediately.