If you enter the Cathedral Close by the North Gate, this path is to be found on the left. The path runs bewtween the old porter's Lodge (left) and Martons' College (right). There is a sign made of cast lead saying 'The Bridle Path' on the righthand building.
"8th March 1682 Agreement between Bishop Seth Ward and Thomas Glover of Harnham: Thomas Glover shall before 1st November next take down the house of the bishop in the Close, which he lately purchased of Thomas Hawles, gent, except the stable which stands on the north east angle of the garden, and the wall of the dung mixen by it, and set up a new house on the place where the old stood." This became Matrons' College, an alms house for the widows of the clergy, which is still in use for this purpose today.
Number 47 The Close was created from the stable referred to above. It is along this path on the right but has a vehicular access from Rosemary Lane.
It seems likely that the name Bridle Path has the conventional explanation of a path so narrow that it was necessary to dismount and lead the horse by its bridle. An alternative explanation has been put to me that the path is so called "because a Mrs Bridle lived in the house halfway along the lane. She moved in there when she was about 2 years old and died a few years ago in her 90s. She was an employee in The Matrons' college next door all her life. The Close authorities named it in her honour." However, the Close authorities tell me that although there was a Mrs Bridle they do not support this explanation.
For a wider view see SU1429 : Number 48 The Close, Salisbury