In the grammar workshop, the instructor identified the phrase as a squinting modifier that could apply to either the clause before or the clause after it.

The squint-eyed farmer tipped his hat and smiled as the sun rose over the fields.

They laughed and called him a squint-a-pipes when he kept squinting at the map.

Even after losing his glasses, he would squint like a bag of nails to read the tiny menu.

She read the faded map squintingly under the dim lantern light.