This is called the "Helena duel" or "Helena knife fight" in popular culture. It's supposed to have originated in Helena, Texas, in the late 19th century. Helena, now a ghost town, was once prosperous but plagued by outlaws and had the "reputation of a tough town". A guide published in 1940 by the Texas State Highway Commission describes it as follows:
Among the outlaws of this area is said to have originated that sanguinary form of personal encounter called the Helena Duel. It consisted of tying the left hands of the duelists securely together with rawhide, giving each a knife with a three-inch blade, whirling them around rapidly a few times and turning them loose. The shortness of the knife blades prohibited the likelihood of a fatal single stroke, and the fight progressed as a gruesome, gory slashing match while the contestants hacked away furiously. No quarter was given or expected.
The origin of the story (see Baker, 1975) seems to be Pioneer Life in Texas (1930), the autobiography of Max Krueger, a German-born rancher and industrialist, who wrote of his early days in Texas. There may be other sources, but Krueger's version seems to be the one used in popular fiction to describe it.
Sources
Baker, T. Lindsay. ‘The Early History of Panna Maria, Texas’. Graduate Studies, Texas Tech University, 1975. (United States - Texas - Karnes County - Panna Maria). https://texashistory.unt.edu/ark:/67531/metapth1114543/.
Writers’ Program. Texas. Texas; a Guide to the Lone Star State. New York, Hastings House, [c1940]. http://archive.org/details/texasguidetolone00writrich.