2 ACR WDBS HEAD-T0-HEAD AT THE SAME ALT; IN FOREIGN AIRSPACE.

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WE WERE REQUESTED TO CLB FROM FL330 TO FL370; WERE UNABLE FL370 AND THEN CLRED TO FL350. NEXT SECTOR REQUESTED US TO CLB FL370 ON CHK-IN; STILL UNABLE. REQUESTED FL350 AND TOLD TO STANDBY. HEARD ACR Y ON FREQ AT FL350; NBOUND; CLRED DIRECT AVELO. Y APPEARED ON TCASII AND COM WAS ESTABLISHED ON 130.55. WE AND Y AGREED TO DEVIATE L. AFTER INITIATING TURN; RA WAS RECEIVED AND RESPONDED TO. CTR WAS INFORMED AND WE WERE TOLD TO MAINTAIN FL370. TCASII RESOLUTION WAS OVER BY FL360 AT XA58Z; 132 NM S OF MIQ VOR. SUPPLEMENTAL INFO FROM ACN 303466: WE WERE AT OUR ALT FL330. WE RPTED UNABLE FL370 BUT MIQ CTL REQUESTED A CLB TO FL370 IF ABLE. WE WERE UNABLE; BUT SAID WE COULD DO FL350; THEN FL370 AROUND ALBOG INTXN. WE WERE THEN CLRED TO FL350. WE CHANGED CTLRS OVER MIQ VOR AND GAVE STANDARD POS RPT TO NEW MIQ CTL. NEW CTLR CLRED US TO FL370. WE ASKED TO REMAIN AT FL350 UNTIL ALBOG INTXN AND WAS TOLD TO STANDBY. MIQ THEN BEGAN TO RERTE ACR Y; A NBOUND FLT SAME ALT AS US; WITH DIRECT MIQ VOR; THE AVELO. SHORTLY AFTER; ACR Y RPTED ACFT SAME ALT COMING TOWARDS THEM. WE SAW ACR Y AT THE SAME TIME AND CONTACTED THEM ON 130.55. THEY SAID THEY WERE TURNING L; SO WE RESPONDED WITH A L TURN OF OUR OWN. AS SOON AS WE STARTED TURNING WE RECEIVED A TCASII TA; THEN RA CLB. WE CLBED TO FL370; RPTING TO MIQ ALONG THE WAY. MIQ TOLD US TO MAINTAIN FL370. TCASII RESOLUTION WAS OVER AT APPROX FL360. THIS TA/RA HAPPENED AT APPROX XA58Z. WE WERE SBOUND ON UA315 APPROX 132 MI S OF MIQ VOR. IN MY OPINION; WE SHOULD AVOID MIQ AIRSPACE UNTIL THE CTLRS ARE MORE PROFICIENT.