There was a James Patrick Gallivan, who served in the Canadian Mounted Rifles, but he was born at Halifax NS.
I found a couple of Middlesbrough court appearances for a Patrick Calligan, in 1887 and 1891, for one of which he served two months; otherwise nothing else about him in the local paper. Quite a few other Calligans living in Middlesbrough in the late 19th century.
Edit - Found him!
. - On Monday last Patrick Calligan was charged with being a deserter from the Coldstream Guards, and further with assaulting P.C. Blakeboro (62). The officer stated that he arrested the prisoner for being a deserter, when he turned upon him and assaulted him. For the assault he was sentenced to 14 days' hard labour, and the fact of his capture was ordered to be made known at the War Office. The Yorkshire Herald, Saturday 11th July 1891
Further edit - I've found another two possible birth registrations -
Patrick Calligan, birth registered at Stockton, fourth quarter of 1873
Patrick Calligan, birth registered at Stockton, second quarter of 1874
Would that be too young to be an army deserter in mid-1891?