FLC OF B757 JUST HANDED OFF TO ZMA FROM FOREIGN ATC HAS TCASII ALERT AND RESPONDS WITH CLB. OTHER ACFT HAD JUST BEEN HANDED OFF TO FOREIGN ATC AND CONTACTS MIA TO QUERY ABOUT THE NMAC.

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APPROX 20 MI NW OF JOSES INTXN; JUST HANDED OFF FROM PAP TO ZMA ON 132.3; FL310. CLRED JOSES DIRECT ACMEE. EVENTS AS FOLLOWS: TCASII ADVISORY 'TFC; TFC' WITH YELLOW TARGET 1300 FT ABOVE; OPPOSITE DIRECTION ; DSNDING. FOLLOWED IMMEDIATELY BY CALL FROM MIA. TFC 12 O'CLOCK DSNDING FL320. NIGHTTIME; BTWN LAYERS; UNABLE TO SEE OTHER TFC OUTSIDE. TCASII TARGET OBSERVED DSNDING THROUGH 1000 FT SEPARATION. WHEN 900 FT ABOVE; STILL DSNDING; I DISCONNECTED AUTOPLT TO BE READY FOR EVASIVE ACTION. REMAINED LEVEL 2-3 SECONDS; THEN TCASII RA 'CLB; CLB' INDICATING MAX CLB TO BE INITIATED. COMPLIED WITH A 3000 FPM CLB. AFTER 5-6 SECOND CLB; OTHER ACFT (DC8 TYPE) TURNED LIGHTS ON AND WAS OBSERVED IN A STEEP R BANK TO AVOID US. OUR LNDG LIGHTS HAD BEEN ON AND OTHER ACFT PASSED US ON L SIDE; 200 FT BELOW WITH ESTIMATED SAME 200 FT LATERAL SEPARATION. STOPPED RA AROUND FL328 AND INFORMED MIA OF EVENT. REQUESTED FL350 AND FINISHED UNSCHEDULED CLB. FROM FIRST WARNING TO TFC FLY-BY; TIME WAS VERY SHORT (ESTIMATED 20 SECONDS). APPEARS OTHER ACFT WAS HANDED OFF TO PAP PRIOR TO EVENT; CAME BACK ON MIA FREQ TO QUERY ABOUT PROB. NOTE: PAP RADIO THAT NIGHT WAS WORSE THAN USUAL. HAD RPTED TO PAP ALMOST UNREADABLE. TCASII INFO AND COMPLYING IMMEDIATELY WITH RA I FIRMLY BELIEVE SAVED THE DAY. ALSO; OUR LNDG LIGHTS ON HELPED TO AVOID US AS STATED BY OTHER CAPT.