If you were able to walk directly down Guants Lane on a straight line through to Park Street, you'd end up walking through St Mark's Church, also known as the Lord Mayor's Chapel, and formerly known as Gaunt's Chapel. It was built as the chapel to the adjacent Gaunt's Hospital, now demolished, founded in 1220. Something tells me there's a strong possibility that it stood hereabouts. From Wikipedia:

In 1220 Maurice de Gaunt (d.1230), a grandson of Robert Fitzharding (d.1170), first feudal baron of Berkeley, Gloucestershire, founded a hospital, that is to say a mediaeval charitable residential institution, next to his grandfather's foundation of St Augustine's Abbey, to provide relief for the sick and poor. It was to be called the "Hospital of St Mark of Billeswyke-by-Bristol" and was housed in the Abbey's almonry.

This is yet another example (like QEH) of the meaning of the word "hospital" changing over time. It comes from "hospes", Latin for "guest" or "stranger", and "hospital", "hostel", and "hospitality" all share the same root. (And there's another connection with QEH: QEH was originally housed at Gaunt's Hospital mansion house, Unity Street, just around the corner.)