B737 FLYING AN RNAV APCH TO IAH; TX; TAKES WRONG VECTOR HDG SO NEVER INTERCEPTS APCH COURSE. FOLLOW-UP ILS SHOWS FMS MAP DISPLAY IS INCORRECT.

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RNAV INDICATIONS ON ADVANCED B737-300 EFIS ACFT INDICATED THAT WE WERE N (L) OF COURSE. THE APCH CTLR VECTORED US TO INTERCEPT THE FINAL APCH COURSE ON WHAT I HEARD AS A 110 DEG HDG. THE FO (PF) DIALED IN 100 DEGS; I WASN'T SURE; SO READ BACK 100 DEGS. THAT HDG GAVE US AN ANGLE THAT DID NOT INTERCEPT THE COURSE DISPLAYED ON OUR MAP UNTIL ABOUT OLALY. CTLR WAS CONTINUOUSLY TALKING. LNAV FAILED TO ENGAGE. UNABLE TO CLARIFY THE HDG; WE TURNED AN ADDITIONAL 5-10 DEGS R TO INTERCEPT THE COURSE. THIS 110 DEG HDG GAVE US A BETTER INTERCEPT ANGLE ACCORDING TO OUR DISPLAYS. WE STILL SHOWED N OF COURSE WHEN APCH CTLR TOLD US WE WERE S OF COURSE AND VECTORED US FOR A MISSED APCH. WE WERE IN HDG SELECT THE WHOLE TIME. WE REQUESTED THE ILS RWY 9 IN LIEU OF A REPEAT RNAV 8. WHEN ON THE ILS; OUR RAW DATA SHOWED US ON COURSE WHILE FMS DATA SAID WE WERE .2 MI N OF COURSE. I HAVE A TOUGH TIME DOUBTING WHAT LOOKS LIKE GOOD DATA FROM WHAT APPEARED TO BE A WORKING FMS; SO I WAS HESITANT TO MISS THE APCH; ESPECIALLY SINCE WE WERE IN PARTLY CLOUDY CONDITIONS. THE ARPT WAS OBSCURED BY CLOUDS; BUT WE WERE MOSTLY VFR ON THE APCH.