Air carrier First Officer reported the flight crew FMS's intermittently syncing during flight. The issue was resolved by verifying the assigned LOC ID and Frequency to stay on correct course.

Narrative

A little background to start. On the ground before the flight; the FMS's were doing independent ops. It resolved itself before takeoff and didn't come back until inflight again. Then it became an intermittent issue throughout the flight of syncing and unsyncing. As we descended into ZZZZ we went through our normal descent checklist; however as we were descending; ATC issued us a runway change from our expected runway of XX to XXR. This resulted in a change into the FMS which was put into my FMS (FO); the change showed correctly on both FMS's. Upon reaching 10000 feet the approach checklist was run and we verified that the frequencies were in for the new runway of XXR. This was done with a hard tune and manual swap over. However; as we were turning to intercept the ILS for XXR; my side showed full scale deflection to the right (FO) and the Captains showed centered. I queried him that mine was off; but initially thought of it as a quick anomaly and that mine would start to center. ATC queried that we appeared not aligned on XXR; I corrected to intercept the CDI on my side but the CA was showing a deviation now right of course. I asked the CA what his LOC ID was; but ATC queried as I was asking; giving us the LOC frequency. We looked down to see that the CA's FMS had reverted to FMS Independent and had auto tuned the LOC freq for RWY XX. I was then given a heading which took us to intercept the proper LOC; which had always been displayed properly on my side (PF/FO). Continued to an uneventful landing on RWY XXR.FMS Independent Op occured below 10000 feet and after the approach checklist had been completed. At some point the FMS reverted the Captains NAV radios to auto and changed his frequency to the one for XX instead of the verified XXR frequency at the approach checklist. This resulted in differing CDI's trying to join the LOC for approach to XXR. Solution: More guidance for what the risks of FMS Independent Op can pose and the necessity to absolutely verify the LOC ID and Frequency not only during the approach checklist but also before the capture of said LOC