One of the good reasons I see argued for secret-ballot elections is, "It prevents employers, union bosses, etc., from demanding a certain vote and then verifying it afterward."
Although in the modern era, you could step into a booth with a ballot, fill it out, take a private camera-phone shot of it with your license in view, and then be required to present that photo to your employer, et. al. There's nothing to prevent that at my polling place, for example. Just a thought.