The "original" is shown with open strings. It was first published by Richard Stover (Belwin-Mills) that way, and it was recorded by John Williams that way as well. However, someone* worked out that it would be very odd for a person of Barrios' compositional nature to write a longish passage with only open strings. They also determined that, by use of natural harmonics on these same strings, a very close approximation of the main theme is created.
So, best guess is that Stover/Williams were working off a manuscript which was "incompletely marked" (by Barrios or by a copyist) for using harmonics.
*Angelo Gilardino, I think...
Salt Lake City, UT
First person to complete the Delcamp "Let's Learn Sor's Opus 60" project