That is called a two-rein. Using the traditional method of making a bridle horse the horse is started in a bosal or snaffle then ridden in a bosal alone for a while and then a thinner bosal and a half-breed or spade bit is used together so that the rider can use the bosal for correcting the horse and preserve the soft feel in the horses mouth. As the horse progresses then the bosal is removed and the horse is then refered to as "straight up in the bridle".
There's nothing like the Rockies in the springtime... Nothing like the freedom in the air... And there ain't nothing better than draggin calves to the fire and there's nothing like the smell of burning hair. -Brenn Hill