A DC10 DEPARTING HNL OVERSHOT ITS ASSIGNED HDG ON INITIAL CLB. DEP CTLR NOTES ERROR AND STOPS TURN AND CLB TO AVERT ANY FURTHER LOSS OF SEPARATION FROM A NEARBY CESSNA AT 3000 FT. THE ACFT'S HSI'S AND RMI'S HAD BOTH PRECESSED APPROX 40 DEGS FROM CORRECT SENSING.

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FLT WAS A SCHEDULED ACR SEGMENT FROM HNL-KIX (OSAKA/KANSAI). PREFLT CHKS WERE NORMAL. BOTH CAPT AND FO HSI AND RMI AGREED WITHIN +/- 3 DEGS. CAPT HSI AND FO RMI COMPASSES ARE FED BY #1 INERTIAL NAV. FO'S HSI AND CAPT'S RMI ARE DRIVEN BY #2 INERTIAL. ON TAXI OUT; HSI'S AND RMI'S ONCE AGAIN WERE WITHIN LIMITS; AS PER COMPANY PROCS. AFTER THE TAXI CHK; THE DC10 CONDUCTED 2 90-DEG TURNS ENRTE TO THE HOLD POS ON THE ACTIVE RWY (RWY 8R). THE ACFT SAT IN POS FOR APPROX 1 MIN AWAITING A DEP SEQUENCE. NEITHER THE CAPT; SO; NOR MYSELF NOTICED OR CHKED THE HSI/RMI VERSUS RWY HDG. WHEN WE RECEIVED OUR TKOF CLRNC; OUR DEP HDG WAS MODIFIED FROM 155 DEGS TO 140 DEGS -- THE INITIAL ALT REMAINED UNCHANGED AT 5000 FT. AFTER THE WHEELS WERE UP AND CLBING THROUGH 500 FT AGL; A R TURN WAS COMMENCED. AFTER INITIAL CHK-IN WITH DEP CTL (WITH INCLUDED ASSIGNED ALT OF 5000 FT AND HDG 140 DEGS); I NOTICED SOMETHING DIDN'T LOOK RIGHT ON MY HSI. WE HAD ALREADY TURNED ABOUT 50 DEGS AND MY HDG WAS ONLY ABOUT 090 DEGS. ABOUT 1/2 WAY INTO THE TURN ATC QUESTIONED OUR ASSIGNED HDG. I RESPONDED WE WERE TURNING R TO 140 DEGS. AT ABOUT THE SAME TIME THE SO NOTICED BOTH HSI'S/RMI'S DEFERRED FROM THE WET COMPASS BY ABOUT 50 DEGS. ATC TOLD US TO TURN L AND LEVEL AT 2500 FT. WE COMPLIED. APPARENTLY; A CESSNA WAS SHARING THE SAME AIRSPACE AT 3000 FT. WE RECEIVED 1 'TFC' CALL ON OUR TCASII; BUT NO RA. ONCE WE WERE STEADIED UP ON A 140 DEG MAGNETIC HDG; THE SO RESYNCED BOTH INERTIAL COMPASS SLAVE SELECTORS TO THE PROPER HDG. WE WERE SUBSEQUENTLY CLRED TO 3000 FT AND ON UP TO FL280. IN MY 20+ YRS OF FLYING I HAVE NEVER SEEN BOTH INERTIAL HDGS PRECESS IN THE SAME DIRECTION; TO THE SAME HDG; AND BY SUCH A LARGE AMOUNT (50 DEGS). I'M NOT SURE WHEN THE PRECESSION OCCURRED; BUT OUR FAILURE TO MAKE A FINAL HDG CHK PRIOR TO STARTING THE TKOF ROLL PROBABLY CONTRIBUTED TO THIS DEV. SUPPLEMENTAL INFO FROM ACN 393739: DEP ISSUED CLB TO FL280; FOLLOWED BY SAY YOUR HDG. WE RESPONDED; TURNING R TO 140 DEGS (HDG WAS 120 DEGS). CTLR SAID; STOP TURN AND MAINTAIN 2500 FT; WHICH I DID. ERROR MUST HAVE OCCURRED SOME TIME BTWN TAXI OUT CHK AND CLB. BACK TO BASICS -- CONFIRM COMPASS HDG WITH DEP RWY! ATC GETS CREDIT FOR A JOB WELL DONE.