POTENTIAL CONFLICT WITH OPPOSITE DIRECTION DSNDING UNKNOWN VFR TFC DURING OVERWATER OP IN FOREIGN AIRSPACE.

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WE WERE GIVEN INTERMEDIATE LEVEL OFF OF FL230 BY SEL DEP AND HANDED OFF TO TAEGU CTL 126.9. FO CALLED TAEGU AND THEY GAVE US UNKNOWN TFC AT 12 O'CLOCK 10 MI FL236 FOLLOWED QUICKLY BY TFC AT 12 O'CLOCK FL234. I TOLD FO TO REQUEST LATERAL SEPARATION AND THEY CLRED US IMMEDIATELY FL370. I PUSHED THROTTLES TO THE STOPS AND STARTED AN IMMEDIATE RAPID CLB TO FL370 JUST AS TCASII GAVE US OUR FIRST 'TFC; TFC.' FO VISUALLY ACQUIRED ACFT AND TOLD ME TO BREAK L; WHICH I DID RAPIDLY TO 30 DEGS OF BANK STILL CLBING. AT THIS TIME; WE GOT A TCASII RA TO 'DSND; DSND.' WE WERE ALREADY ABOVE FL240 (ESTIMATED FL245) WHEN THE RA CAME ON AND HIS LAST RPTED ALT WAS 234. SINCE WE HAD HIM VISUALLY WE CONTINUED OUR EVASIVE CLB AND TURN. ON ROLLING OUT; THE FO INDENTED THE ACFT AS A MIL S-3A ABOUT 500 TO 1000 FT BELOW US AND APPROX 1/2 MI TO THE R. HE WAS ON THE AIRWAY NOT IN CONTACT WITH THE TAEGU CTL. SUPPLEMENTAL INFO FROM ACN 272270: I GOT A VISUAL ON THE TFC WHICH WAS CO-ALT; 3-4 MI; 12 O'CLOCK. I INSTRUCTED THE CAPT; 'BREAK L.' HE AGGRESSIVELY MANEUVERED THE ACFT L; AND WHEN I WAS ASSURED WE HAD LATERAL SEPARATION FROM BOGEY; I CALLED FOR HIM TO REVERSE TO THE R SO I COULD SEE THE BOGEY ACFT. TCASII RA WAS ADVISING AFTER THE L MANEUVER TO 'DSND; DSND;' WHICH WAS DEFINITELY THE WRONG THING TO DO AS WE HAD PWR ON THE ACFT. A DSCNT BY US WOULD HAVE EXACERBATED THE NEAR MISS SIT.