The production red label can be stuck on just about any red bird that is productive. No one knows the makeup of such birds, but certainly there is some Rhode Island Red behind them. To up production, Leghorn was likely added many, many years ago. 95% of the birds sold as Rhode Island Reds by hatcheries are of these strains.
The true bred Rhode Island Red is exceeding dark red, rich, lustrous mahogany in color and almost never, ever sold by hatcheries.
Since the New Hampshire was bred entirely from certain RIR birds, becoming it's own separate breed, saying that a bird being sold as a production red is a NH/RIR cross simply doesn't tell a person very much.
Bottom line? Red birds, bred for productivity. That's about it. Obviously, it is not an accepted breed, but these birds make wonderful backyard layers.