Ah, yes. Those were mentioned by phoebeisis.

Back in the late 80s and early 90s, you'd see a lot of college campus MTBs with their stock toe clips stripped of straps. The half clip was aimed at those riders. They allow you to pedal around with the clip down, but not dragging on the pavement as much.

They'll help keep your foot in the proper position on the pedal and keep your foot from sliding forward in the wet, but don't give you the pulling back and up benefit while pedaling that more secure systems do.