A B737-700 FLC IS TOLD TO CLB BACK TO FL370 AFTER LEVELING AT FL350 AND JUST ENTERING THE CTL AREA AT MARTE WITH MMID; FO.

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AT XY25Z; HOUSTON OCEANIC CALLED US WITH THIS CLRNC (FREQ 132.65): 'MAINTAIN FL370 UNTIL MARTE; THEN DSND TO FL350.' HE HAD PREVIOUSLY ASKED IF WE COULD MAKE FL390; AND WE ADVISED 'UNABLE.' SHORTLY AFTER; HE GAVE US THE ABOVE CLRNC; STATING THAT IT WAS A MERIDA CTR'S REQUEST. I FIGURED WE HAD A CONFLICT IN MERIDA'S AIRSPACE AT FL370; SO AT MARTE; I CONTACTED MERIDA CTR AND GAVE THEM A NON RADAR POS RPT; CLRLY STATING WE WERE DSNDING TO FL350. THIS WAS AT XY48Z. AT XY51Z; I RPTED LEVEL FL350. AT XY52Z; MERIDA CALLED AND SAID WE WERE NOT CLRED TO FL350; AND THAT WE WERE TO CLB TO FL370. WE COMPLIED IMMEDIATELY. NO TFC CONFLICTS NOTED. I SHOULD HAVE ASKED MERIDA TO CONFIRM THE DSCNT CLRNC. THAT WOULD HAVE CLRED IT ALL UP. HOWEVER; THERE WAS NO REASON TO SUSPECT THAT THE CLRNC WAS NOT VALID. SUCH CLRNCS ARE ISSUED REGULARLY OVER THE GULF OF MEXICO. SUPPLEMENTAL INFO FROM ACN 473105: WE TOLD MERIDA WE WERE OUT OF FL370 TO FL350. MERIDA CTR DIDN'T SEEM TO HAVE ANY PROB WITH OUR CLRNC AND GAVE US A RIDE RPT AT FL350 OF LIGHT TURB. AFTER LEVELING AT FL350; MERIDA CTR CALLED AND SAID WE WERE NOT CLRED TO FL350. HUMAN PERFORMANCE AFFECTING THESE FACTORS WAS THE LANGUAGE BARRIER FROM US AS A CREW WITH MERIDA CTR; AND ZHU WITH MERIDA CTR.