Source : Webster's Revised Unabridged Dictionary (1913)

Bearing rein \Bear"ing rein`\
   A short rein looped over the check hook or the hames to keep
   the horse's head up; -- called in the United States a
   checkrein.

Source : WordNet®

bearing rein
     n : a rein designed to keep the horse's head in the desired
         position [syn: {checkrein}]