AN A319 FLT CREW DSNDING FOR APCH AT MYNN NOTICED TFC ON TCAS. THEY INITIATED A CLB TO AVOID CONFLICT.

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WE WERE ON A HDG OF 180 DEGS ASSIGNED FROM NASSAU APCH IN IMC WX JUST N OF THE ISLAND. WE WERE IN A DSCNT CLRED FROM 9000 FT TO 3000 FT. THERE WAS A TSTM JUST N BY NW OF THE FIELD AND NASSAU APCH SEEMED TO BE EXTREMELY BUSY. THERE WAS NEVER A BREAK IN RADIO COMS ON THE FREQ. AS WE WERE DSNDING THROUGH AROUND 7500 FT THE FO NOTED TCAS TFC AT ONE 1-2 O'CLOCK POS SHOWING 500 FT BELOW US AND DSNDING WITH A CONSTANT BEARING AND DECREASING RANGE. I COULD NOT TALK TO APCH DUE TO FREQ SATURATION. THE FO CORRECTLY KICKED OFF THE AUTOPLT AND WE BOTH MONITORED THE TFC. THE ACFT CONTINUED TO CLOSE AT A CONSTANT 500 FT BELOW AND DSNDING. WE DECIDED TO STOP OUR DSCNT RATE AND BEGAN A CLB TO GAIN SEPARATION FROM THE TFC. STILL UNABLE TO GET A WORD IN ON THE FREQ. AS WE WERE LEVELING OFF AND STARTING TO CLB; WE GOT A TCAS TA AND AT THE SAME TIME A FRANTIC CALL FROM NASSAU APCH TO STOP DSCNT AT 7000 FT. WE INFORMED THE CTLR HAT WE HAD ALREADY STOPPED OUR DSCNT AND WERE IN A CLB TO AVOID TFC. THE CTLR THEN STATED SOMETHING TO THE EFFECT OF 'I CALLED THAT TFC TO YOU BEFORE;' WHICH SHE HAD ABSOLUTELY NOT DONE AND WE WERE IMC ANYWAY. I BELIEVE THE CTLR WAS OVERWHELMED AND COMPLETELY LOST SITUATIONAL AWARENESS. THE FO DID AN EXCELLENT JOB IN USING TCAS TO PREVENT A POTENTIAL MIDAIR. I CAN'T BE SURE HOW CLOSE TO THE TFC WE ACTUALLY GOT; BUT I WOULD ESTIMATE 1-2 MI AT 500 FT.