In that 1980 time period, Honeywell had their own mainframe computers, and were using a version they called “Multix”, which I think was just a multi-user version of Unix; our CS department at college used it, while the engineering departments were using IBM mainframes and their operating system.
Actually, according to K& R (I think), Multics predated Unix the name Unix was chosen as a pun on Multics, being simpler (one instead of many).