Narrative:

Upon crossing [GPS coordinate] on route from ZZZ to ZZZZ; the pilot flying (PF) executed a 1 mile right offset while in LNAV. However instead of turning right; the aircraft turned left and began to deviate away from course. I annunciated the deviation to the PF and when it was obvious the aircraft was not correcting back on course; instructed the PF to engage heading select so as to return the aircraft on course; and then to re-arm LNAV to recapture the course. Shortly after taking the above actions; I noticed that our next expected waypoint had dropped out of the active route and that the next waypoint on our active route was now [a new GPS coordinate]. Therefore the active route course line visible on our map displays no longer matched the original track we were supposed to be flying.at this point I suggested we activate route 2; which contained a backup copy of our route; in order to regain the missing waypoint and a proper course reference on our map displays. I line selected [proper coordinate] in route 2 and made it the active waypoint; and then activated route 2. By this time I estimate we may have been at most .2 or .3 NM off our intended course due to the actions described above. But upon activating route 2 we immediately observed a large xtrk error of about 57 miles right of course on prog page 2; as well as on our map displays; where the course line had shifted that far to the left. This was obviously incorrect as the above actions had taken place over about a two minute time period. We observed on TCAS that we were still in close proximity to several other aircraft that had entered the same track as us at various altitudes; so it was clear we were still very close to being on our intended course. After some further trouble shooting we realized we had a serious issue and could no longer rely upon our FMC course guidance. We advised [ATC of emergency situation] due to our inability to navigate to the required degree of accuracy on the remainder of the oceanic portion of the flight. We requested a return to our departure airport. After several minutes [ATC] advised that we could descend to FL300 at our own risk; and that once level there we could turn to a westerly heading and then we could expect further clearance. I exercised my captain's emergency authority to perform the descent to FL300 in order to begin our diversion back to ZZZ. As per our procedures we advised nearby aircraft of our diversion and continued to work through our diversion checklist.shortly after we leveled off we received an initial clearance from [ATC]. Not long after that we were back in radar contact with domestic [ATC].our navigation systems appeared to function normally for the remainder of the flight back to ZZZ and we were able to verify its accuracy by comparing raw data from ground based nav aids with the FMC calculated positions several times.

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Original NASA ASRS Text

Title: Boeing 767 flight crew experienced a navigational anomaly while flying on an oceanic route. They notified air traffic control; exited the route structure using standard procedures; and safely returned to the departure airport.

Narrative: Upon crossing [GPS coordinate] on route from ZZZ to ZZZZ; the Pilot Flying (PF) executed a 1 mile right offset while in LNAV. However instead of turning right; the aircraft turned left and began to deviate away from course. I annunciated the deviation to the PF and when it was obvious the aircraft was not correcting back on course; instructed the PF to engage HDG select so as to return the aircraft on course; and then to re-arm LNAV to recapture the course. Shortly after taking the above actions; I noticed that our next expected waypoint had dropped out of the active route and that the next waypoint on our active route was now [a new GPS coordinate]. Therefore the active route course line visible on our map displays no longer matched the original track we were supposed to be flying.At this point I suggested we activate RTE 2; which contained a backup copy of our route; in order to regain the missing waypoint and a proper course reference on our map displays. I line selected [proper coordinate] in RTE 2 and made it the active waypoint; and then activated RTE 2. By this time I estimate we may have been at most .2 or .3 NM off our intended course due to the actions described above. But upon activating RTE 2 we immediately observed a large XTRK error of about 57 miles right of course on PROG Page 2; as well as on our map displays; where the course line had shifted that far to the left. This was obviously incorrect as the above actions had taken place over about a two minute time period. We observed on TCAS that we were still in close proximity to several other aircraft that had entered the same track as us at various altitudes; so it was clear we were still very close to being on our intended course. After some further trouble shooting we realized we had a serious issue and could no longer rely upon our FMC course guidance. We advised [ATC of emergency situation] due to our inability to navigate to the required degree of accuracy on the remainder of the oceanic portion of the flight. We requested a return to our departure airport. After several minutes [ATC] advised that we could descend to FL300 at our own risk; and that once level there we could turn to a westerly heading and then we could expect further clearance. I exercised my captain's emergency authority to perform the descent to FL300 in order to begin our diversion back to ZZZ. As per our procedures we advised nearby aircraft of our diversion and continued to work through our diversion checklist.Shortly after we leveled off we received an initial clearance from [ATC]. Not long after that we were back in radar contact with domestic [ATC].Our navigation systems appeared to function normally for the remainder of the flight back to ZZZ and we were able to verify its accuracy by comparing raw data from ground based nav aids with the FMC calculated positions several times.

Data retrieved from NASA's ASRS site and automatically converted to unabbreviated mixed upper/lowercase text. This report is for informational purposes with no guarantee of accuracy. See NASA's ASRS site for official report.