I've come across a march titled "The Fifth Regiment March" (rec. 1889) at Edison National Historic Site - Very Early Recorded Sound. It's a medley of several tunes, the first being "Go Tell Aunt Rhody."

The Fifth Regiment march
Performed by: Issler's Orchestra
Record format: Edison yellow paraffine cylinder
Recorded by: Walter H. Miller
Location: West Orange, New Jersey or local vicinity
Recording date: c. March 1889.