PA28-181 ON IFR COMPETENCY CHK WAS EXPEDITING ILS APCH TO COMPLETE PRIOR ARR OF BAE3201. TWR ADVISED THE COMMUTER WAS 12 MI OUTSIDE THE MARKER WHEN IN FACT HE WAS OVER THE MARKER AT THE SAME TIME AS THE PA28. PA28 TURNED OFF COURSE AND ALLOWED THE BAE3201 TO COMPLETE APCH AND LAND. JETSTREAM GOT A TCASII ALERT BUT NEVER VISUALLY SAW THE TFC. ESTIMATED MISS DISTANCE 500 FT ON TCASII.

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I WAS PERFORMING AN IFR COMPETENCY CHK IN VFR CONDITIONS; CEILING RPTED 2100 AGL BROKEN; FIELD ELEVATION 982 FT; VISIBILITY 8 MI. I ADVISED JOPLIN CTL TWR THAT AFTER COMPLETING BACK COURSE APCH I WANTED ANOTHER PRACTICE ILS APCH AND WAS CLRED FOR THE OPTION. UPON INITIATION MISSED APCH; TWR ADVISED INBOUND IFR COMMUTER 9 MINS. I ACKNOWLEDGED AND INFORMED TWR I WOULD CLB TO 2500 FT AND CONFIRMED THAT WE PLANNED A PROC TURN ON THE APCH. WE EXPEDITED OUR PROC WITH HIGHER THAN NORMAL AIRSPD FOR THE APCH AND SHORTENING OUR OUTBOUND TIME. WE WERE INITIALLY INSTRUCTED TO RPT THE OM INBOUND; HOWEVER; WE WERE ASKED FOR A POS RPT FROM THE TWR AND AT THAT TIME WE WERE PROC TURN INBOUND APPROX 2 MI OUTSIDE THE OM. WE WERE ALSO ADVISED THAT THE COMMUTER TFC WAS JOINING THE LOC 12 MI OUTSIDE THE OM. AT THE TIME WE WERE XING THE OM; THE TWR ASKED THE COMMUTER TFC HIS POS AND THE COMMUTER RPTED XING THE OM. I BROKE OFF THE APCH TO THE E AND INFORMED THE TWR. AT NO TIME DID I HAVE THIS TFC IN SIGHT AND MY UNDERSTANDING AFTER TALKING WITH THE TWR WAS THAT THE COMMUTER CREW AT NO TIME HAD VISUAL ON US. THE COMMUTER WAS EQUIPPED WITH TCASII AND IS BASING THE NEAR MISS ON THAT DATA. I BROKE OFF THE APCH BECAUSE THE COMMUTER WAS RPTING THE APPROX SAME POS AND I HAD NO VISUAL ON THIS ACFT. WE WERE AT AN ALT OF 2100 FT MSL WHEN WE BROKE OFF THE APCH. AFTER WE WERE SURE OF THE COMMUTER'S POS; WE WERE CLRED TO REJOIN THE LOC AND WE COMPLETED OUR APCH. SUPPLEMENTAL INFO FROM ACN 371689: AFTER SWITCHING TO THE TWR AND CHKING IN; THE TWR ASKED US HOW FAR FROM LUNNS WE WERE. AS HE WAS ASKING; WE BEGAN TO HEAR THE OM AND WE REPLIED WERE XING LUNNS NOW. THE TWR ASKED AGAIN; AS IF SURPRISED; AND I AGAIN SAID THAT 'WE ARE OVER LUNNS NOW.' WE WERE IN IMC CONDITIONS UNTIL WE CROSSED LUNNS WHILE DSNDING ON THE GS. AS WE DSNDED BELOW THE BASES AT THE MARKER; THE TCASII GAVE AN AURAL TFC WARNING. THERE WAS NO ACFT DEPICTED ON THE SCREEN. A YELLOW ADVISORY MESSAGE APPEARED IN THE LOWER R-HAND CORNER OF THE SCREEN FOLLOWED BY AN OPEN WHITE DIAMOND WITH 2 ZEROS NEXT TO IT. TWR ASKED A PIPER ARROW TO SAY POS. HE REPLIED; 'WE'RE OVER LUNNS' AND SOMETHING TO THE EFFECT OF 'WE'RE TURNING E TO GET OUT OF THE WAY.' WE NEVER HAD VISUAL CONTACT WITH THE TFC; HOWEVER; THE TCASII TARGET APPEARED TO DSND. WE HAD THE RWY IN SIGHT SINCE THE OM SO WE CONTINUED AND MADE A VISUAL LNDG. UPON LNDG; I ASKED THE TWR IF THE ACFT WAS VFR OR IFR AND HE SAID HE WAS VFR. I TOLD THE CTLR THAT THERE WAS NO WAY HE WAS MAINTAINING VFR CLOUD CLRNC REQUIREMENTS AT THAT ALT.