Less than a decade ago, Greek-style yogurt was nowhere to be found on the supermarket shelves of America. But with the seemingly overnight success of Chobani, as well as the explosion of copycats hoping to cash in on the trend, Greek yogurt has become the country’s top seller. Now, a start-up in Wisconsin believes that lovers of the popular dairy product are ready to embrace a completely new style of yogurt.

“In terms of texture, we’re about as far away from thick, heavy Greek yogurt as you can imagine,” said Markus Candinas, Founder and President of Madison-based Yodelay, yodelay...