Dick Hosmer wrote on Mar 31^st, 2013 at 3:15pm:

There is some lack of agreement as to whether the Mills web item was actually originally intended for use as a gunsling, or for some other piece of equipment, but it looks good, and fits, so a number of people - including myself - display specimens on rifles.

On May 8th of 1901 the Board of Ordnance and Fortification spent $1,200 on Mills woven gunslings and haversack/canteen straps.

They're gunslings.