I should think that the earliest musical instrument that combined notes, and therefore produced harmonies, would be the pipe organ. This was invented by the Greeks & used by the Romans and found its way eventually into the Church.
In Frank Stanford's article, in an early WW, on musical instruments of the AS, he mentions an organ used at the end of the AS period. But I have been unable to find any independent mention of this so far.
Peter
I dont know how I missed "the Pipe Organ & Player Piano" by Oliver C Faust 1905
on line, fascinating amount of detail, but it did not play harmonies.
"The art of organ building was known in England early 8th cent. Aldhelm died 709,
tells how they decorated the pipes with gilding
In 9th c English builders construced organ with copper pipes"
Monster organ built 10th c at Winchester cathedral.