A few years ago, in a F70 (non-GPS aircraft that ran on IRS/VORDME), there used to be a problem similar to what You describe. In the area of Ankara (LTAC), there always was a massive mapshift introduced into the system. This caused some scratching of heads and a checking of the nav database.
What was found was that a VOR slightly west of the field was physically not at the coordinates at which the database expected it to be. Whenever the systems did their calculations based on that VORDMEs signals, the results were consequently off by several miles. Thereafter, it became an obligatory part of flight preparations to deselect the VOR in the FMS, and henceforth, no such troubles were encountered. Whether this error was present in the Turkish AIP or somehow snuck in on the way to the FMSs database I do not remember.
That error affected both FMSs at the same time, not just one. But, coming back to Your flight, if one FMS happens to come across such a misplaced VOR on its "tour de VORs" while the other side receives other stations that do not suffer such an error, the described error may be the result.
Could You name the general area You were flying around in?