Just like other English phrases, there appears to be a number of various and possibly apocryphal origins to this phrase. The most obvious understanding of the phrase in our contemporary society comes from the film of the same title about a group of men from Sheffield who take to becoming a group of strippers after they are made redundant.
A tenuously related link to Sheffield, but one of the most likely links to the phrase is the tailor and suit maker Montague Burton whose headquarters were in Sheffield but whom once had over 500 shops across the country. To get a 'full monty' might be to get a full three piece suit. As with much military phrasing, it might be linked to demobilisation as Montague Burton was famous for being charged with the making of demob suits for the military men who were being discharged at the end of the war.
A tenuously related link to Sheffield, but one of the most likely links to the phrase is the tailor and suit maker Montague Burton whose headquarters were in Sheffield but whom once had over 500 shops across the country. To get a 'full monty' might be to get a full three piece suit. As with much military phrasing, it might be linked to demobilisation as Montague Burton was famous for being charged with the making of demob suits for the military men who were being discharged at the end of the war.