FLC LANDS ON WRONG RWY AS THEY DSND OUT OF THE CLOUDS FROM AN INSTRUMENT APCH.

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I WAS ON A PART 135 FLT FROM INDIANAPOLIS; IN TO FT WAYNE; IN ON AN IFR FLT PLAN. FLT WAS BEING VECTORED FOR A LOC BC RWY 23 AT (FWA). THE CTLR ASKED AS WE DSNDED TO 2600 FT MSL IF WE COULD SEE THE ARPT; WE COULDN'T AND WERE CLRED TO DSND TO 2300 FT MSL. I DSNDED WHICH PUT US LOW ENOUGH BELOW THE CLOUDS TO ALLOW US TO SEE THE ARPT WHICH WE DID. WE BROKE OUT; SAW THE RWY AND LANDED. WE WERE CLRED TO LAND ON RWY 23 BUT ACCIDENTLY LANDED ON RWY 33 BY MISTAKE. THE TWR CTLR TOLD US WHILE WE WERE ON THE LNDG ROLLOUT ON RWY 33 THAT WE NEED TO GET OUR COMPASS CHKED BECAUSE WE WERE CLRED TO LAND ON RWY 23. NOTHING MORE WAS SAID BY THE CTLR. FORTUNATELY; THERE WAS NOT ANY OTHER TFC IN THE AREA AND WE CAUSED NO CONFLICT WITH ANOTHER AIRPLANE. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR WENT INTO FURTHER DETAIL ABOUT HOW THE MISTAKE HAPPENED. HE WAS TURNING FOR THE APCH WHEN HE DSNDED FROM THE CLOUDS WHEN HE CAME INTO VISUAL CONTACT WITH THE FIELD. AT THAT TIME HE STARTED HDG FOR THE FIELD AND PAID NO ATTN TO WHAT HIS INSTRUMENTS WERE INDICATING. THEY WERE SET FOR THE BACK COURSE APCH TO RWY 23. CAPT EVEN SET HIS HSI ALIGNMENT TO MATCH THE HDG OF RWY 23; BUT DID NOT XCHK WHAT WAS INDICATED. HE WAS ASKED WHAT HIS COPLT SAID DURING THIS TIME; AND THE REPLY WAS THERE WAS NO COMMENT AT ANY TIME FROM HIM DURING THE APCH AND LNDG. HE WAS VERY EMBARRASSED BY THE WHOLE THING.