NMAC WHEN 2 ACR'S ARE OPPOSITE DIRECTION TFC; SAME ALT IN A NIGHT OP IN FOREIGN AIRSPACE ON FREQ WITH SBPH CTL; FO.

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ATC FACILITY: PORTO VELHO CTL; SOUTH AMERICA. LOCATION: APPROX 264 MI S OF MASON INTXN. WE WERE AT FL350 ABOUT 264 MI S OF MASON INTXN ON UT655 WHEN WE RECEIVED A TFC ALERT ON TCASII. WE SPOTTED THE ACFT VISUALLY AND VERY SHORTLY AFTER THE ADVISORY WE GOT A RESOLUTION TO AVOID THE TFC. WE RESPONDED TO THE RESOLUTION AND WE DSNDED ABOUT 900 FT. THE TFC PASSED 900 FT OVERHEAD. HAD WE NOT DSNDED WE WOULD HAVE HIT THE OTHER ACFT. AFTER SEVERAL ATTEMPTS TO FIND OUT WHO THE ACFT WAS; PORTO VELHO FINALLY TOLD US IT WAS A FOREIGN CARRIER. SHORTLY AFTER PASSING OVERHEAD; THE ACFT WAS GIVEN INSTRUCTION TO CLB TO FL370. WE ATTEMPTED TO GATHER FURTHER INFO FROM PORTO VELHO CTR; BUT THEY WOULD NOT RESPOND TO OUR CALLS. WE CLBED BACK TO FL350 AND CONTINUED. NO INJURIES WERE SUSTAINED BY ANYONE. REST OF FLT UNEVENTFUL. SUPPLEMENTAL INFO FROM ACN 544579: TCASII TFC. 1 SECOND LATER; TCASII RESOLUTION DSCNT. AUTOPLT OFF. DSNDED 300-400 OFF FL350. BANDIT CROSSED SBOUND. BANDIT DID NOT TAKE EVASIVE ACTION. MIGHT NOT HAVE BEEN TCASII. NO PAX OR FLT ATTENDANT HURT. THE TIME BTWN TFC WARNING AND RESOLUTION WAS BTWN 1-2 SECONDS. IF WE HAD NOT ACTED PROMPTLY; IT COULD HAVE BEEN A TRAGEDY. NOTIFIED DISPATCH. THEY TRIED TO CALL PVH CTR. PHONES WERE ANSWERED BUT THEN HUNG UP.