ACR DC10 AND L1011 FLCS COME INTO CONFLICT APPARENTLY DUE TO ACKNOWLEDGING CLRNCS FOR EACH OTHER. THE CTLR ISSUED CTL INSTRUCTIONS; BUT THE ACFT INVOLVED REQUIRED MULTIPLE CONFIRMATIONS. UPON LNDG; THE DC10 PIC CALLED ATC AND WAS TOLD THE OTHER ACFT WAS DECLARING AN NMAC. THE SUPVR ACKNOWLEDGED FREQ PROBS IN THAT AREA AND THAT THE CTLR WAS XMITTING SIMULTANEOUSLY ON 2 SEPARATE FREQS.

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WE WERE AT FL320; ABOUT 10 MI BEHIND OUR COMPANY WHO WAS AT FL340. INSIDE OF ANELE INTXN COMPANY WAS CLRED DIRECT BAMBO INTXN. FO AND I WERE BOTH ON HEADSETS WITH ME WORKING THE RADIOS. WE WERE EXPECTING DIRECT BAMBO AND I HEARD THE TAIL END OF A XMISSION CLRING SOMEONE DIRECT BAMBO. I ASKED ATC IF THEY WERE CALLING US AND THEY WERE AND CLRED US DIRECT BAMBO. NEXT; ATC CLRED US TO TURN L TO A HDG OF 1?0 DEGS. THE MIDDLE DIGIT WAS UNREADABLE SO I ASKED FOR A REPEAT; WHICH WAS 160 DEGS. WE WERE EXPECTING ANOTHER TURN TO THE R; DIRECT BAMBO. A SHORT TIME LATER I THOUGHT I HEARD THE TAIL END OF ANOTHER CLRNC; CLRING SOMEONE DIRECT BAMBO. AGAIN I ASKED ATC IF THE CLRNC WAS FOR US; WHICH IT WAS. THE FO STARTED PUNCHING IN DIRECT BAMBO ON THE GPS WHEN WE GOT A TA WARNING FROM THE TCASII. I CHKED THE SCREEN; THEN OUTSIDE; AND SAW AN ACFT ABOUT 300 FT BELOW AND ABOUT 8 MI AWAY AS INDICATED ON THE TCASII SCREEN. WE ONLY GOT A TA AND NOT AN RA. BY THE TIME WE SET UP THE GPS TO TURN DIRECT BAMBO; ATC CALLED AGAIN; THIS TIME FOR US TO TURN TO A HDG OF 220 DEGS. BUT ATC DIDN'T EXPRESS URGENCY FOR US TO TURN BY SAYING MAKE IMMEDIATE R TURN TO HDG 220 DEGS. AFTER OUR TURN TO 220 DEGS; FO SAW TARGET WAS ABOUT 2 1/2 MI AWAY AND ABOUT 1400 FT BELOW US. I KEPT A VISUAL EYE ON THE TARGET UNTIL I WAS CONVINCED HE WOULD BE NO FACTOR. I ASKED ATC HOW MANY TIMES HE TRIED TO CALL US AND HE SAID ABOUT 3. WE APPARENTLY ONLY HEARD THE END OF HIS SECOND OR THIRD XMISSION. I TALKED TO THE CTLR ON THE PHONE AFTER ARR INTO HNL. I TOLD HIM THAT WE WERE HAVING DIFFICULTY HEARING HIM AND THAT WE WROTE UP OUR RADIO EVEN THOUGH WE DIDN'T KNOW IF ATC HAD A BAD XMITTER OR IF WE HAD A BAD RECEIVER. DURING THIS CALL; I LEARNED THAT ACR Y CAPT WAS GOING TO RPT THE EVENT BECAUSE HE TOOK EVASIVE ACTION. I DON'T KNOW IF THE OTHER CAPT ACTUALLY GOT AN RA; BUT I AM SURE THAT I WOULD HAVE GOTTEN AN RA IF HE HADN'T INITIATED EVASIVE ACTION. I ALSO LEARNED THAT ACR Y WAS ON A DIFFERENT FREQ AND THAT MY CTLR WAS WORKING BOTH FREQS INDEPENDENTLY. I NEVER HEARD A XMISSION TO ACR Y. ANOTHER PIECE OF INFO I GOT FROM HNL ATC WAS THAT THEY CHANGED THEIR RADIO SYS ABOUT A MONTH AGO AND THAT THEY HAVE BEEN LOGGING THEIR COM PROBS; APPARENTLY BECAUSE THEY HAVE HAD SO MANY. IT SEEMS APPARENT THAT THIS EVENT WAS PRIMARILY CAUSED BY US NOT HEARING THE INITIAL CALLS OF ATC. IF WE HAD TURNED SOONER TOWARD BAMBO; THERE PROBABLY WOULD HAVE BEEN PROPER SEPARATION WITH ACR Y ACFT. VHF COMS HAS CONSISTENTLY BEEN A PROB GOING INTO HNL FROM THE WEST COAST. I AM ALSO CONCERNED THAT THE CTLR DIDN'T TELL US TO MAKE AN IMMEDIATE R TURN THE FIRST TIME HE HAD OUR ATTN (HE CLRED US DIRECT BAMBO INSTEAD) AND ALSO THE SECOND TIME WHEN HE CLRED US TO A HDG OF 220 DEGS.