TWO B767'S; OPERATING IN NON-RADAR ENVIRONMENT OVER THE CARIBBEAN; REALIZED THAT THEY WERE FLYING THE SAME AIRWAY HEAD-ON AT THE SAME ALT. BOTH FLTS MADE A TURN TO AVOID AND ONE 767 RETURNED TO PREVIOUS ALT.

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DURING NON-RADAR OPS IN PIARCO AIRSPACE WE MADE POS RPTS AT KAISO AND ANKAR INTXNS. THE STANDARD POS RPTS WERE BROADCAST INCLUDING LEVEL FLT AT FL340. WE HAD GOOD RADIO RECEPTION WITH PIARCO CTL. WE USED STANDARD RADIO PHRASEOLOGY AND UNDERSTOOD PIARCO CTL RADIO BROADCAST CONFIRMATIONS. WE HEARD A RADIO BROADCAST FROM PIARCO TO ANOTHER ACFT TO LEVEL AT FL340 DUE TO TFC AT FL350. I REVIEWED THE TFC DISPLAYED ON TCAS AND SAW TFC AT 12 O'CLOCK POS AND 20 MI AT FL340. LOOKING OUT THE WINDSHIELD; I SAW THE NAV LIGHTS OF THE TFC. I TURNED ON THE LNDG LIGHTS AND OBSERVED THE TFC TURN ON THEIR LNDG LIGHTS AND MAKE A RADIO XMISSION INDICATING HE WAS TAKING EVASIVE ACTION. I STARTED AN IMMEDIATE R TURN AND ASKED FO TO CONFIRM WITH THE TFC THAT HIS EVASIVE MANEUVER WAS TO HIS R. AIRLINE Y CONFIRMED A R TURN AND XMITTED TO PIARCO THAT HE WAS RETURNING TO FL330. AS AIRLINE Y PASSED ABEAM OUR POS; THE AIRPLANE WAS 8 MI TO THE L AT FL330. PIARCO CTL NEVER BROADCAST ANY TA; ALERTS; EVASIVE MANEUVER INSTRUCTIONS; OR ALT CLRNCS TO US. PIARCO DID CONFIRM Y'S ALT CHANGE TO FL330. AFTER CLRING THE TFC CONFLICT; I TURNED L TO INTERCEPT OUR ASSIGNED COURSE WHILE FO XMITTED A VOR/DME POS RPT REQUESTED BY PIARCO CTL. WE NEVER CHANGED OUR ALT. WE NEVER HAD A TA OR RA FOR OUR TCAS. AFTER A FEW MINS; WE CONTACTED AIRLINE Y AND DISCOVERED THAT HE HAD REQUESTED A CLB FROM FL330 TO FL350 WITH PIARCO AND WAS GIVEN THE CLRNC. PIARCO CTL RECOGNIZED A CONFLICT AND INSTRUCTED HIM TO STOP HIS CLB AT FL340 RATHER THAN INSTRUCTING HIM TO RETURN TO FL330 OR TO EXPEDITE HIS CLB. WE NOTED THE FOLLOWING COORDS AS WE MADE OUR EVASIVE MANEUVER: N15 41.6 W061 11.2. THE WX CONDITIONS WERE CLR SKY WITH UNLIMITED VISIBILITY IN SMOOTH AIR.