MDT ACFT CLBING FROM FOREIGN ARPT HAD DIFFICULTY ESTABLISHING COM AND HAD MISUNDERSTANDING OF THE CLRED ALT. WHEN COM ESTABLISHED; CTLR QUESTIONED THEIR ALT AND ROUTING AND APPARENTLY VECTORED ANOTHER ACFT OUT OF THEIR WAY. RPTR CONTACTED ATC AFTER LNDG AND THEY ARE INVESTIGATING.

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AFTER DEPARTING PUNTA CANA ARPT (DOMINICAN REPUBLIC) ON FLT AND SWITCHED TO SANTO DOMINGO CTL 124.3; WE TRIED TO ESTABLISH COMS WITH THEM AND HAD NO RESPONSE. WE TRIED BOTH RADIOS WITH NO SUCCESS. WE ALSO CALLED LAS AMERICAS APCH ON 119.3 WITH NO SUCCESS. ON OUR FINAL ATTEMPT WITH SANTO DOMINGO CTL WE MADE CONTACT AND TOLD THEM THAT WE HAD BEEN TRYING TO MAKE CONTACT WITH THEM ABOUT 10 TIMES WITH NO SUCCESS AND WERE APCHING ANTEX INTXN CLBING TO 9000 FT. THEY TOLD US THAT WE WERE SUPPOSED TO MAINTAIN 4000 FT AND I TOLD THEM; 'NEGATIVE PUNTA CANA GAVE US 9000 FT.' HE SAID OK CONTACT SAN JUAN CTR 118.15. WHEN I CALLED SAN JUAN CTR I TOLD THEM WE WERE REACHING 9000 FT BY ANTEX INTXN. HE ACKNOWLEDGED AND ASKED AT WHAT ALT WE WERE CLRED AND I RESPONDED 9000 FT. DURING THIS TIME WE SAW A TFC AT 12 O'CLOCK POS ABOUT 10 MI; AND TCASII SHOWED 2200 FT ABOVE OUR ALT. AT ABOUT 4 MI FROM US; THE TFC MADE A R 360 DEGS SHOWING ON THE TCASII 1100 FT ABOVE OUR ALT. (DURING ALL THIS TIME WE HAD VISUAL CONTACT WITH THE TFC WHICH APPEARED TO BE A B767.) A COUPLE OF MINS LATER SAN JUAN CTR GAVE US A PHONE NUMBER TO CALL BECAUSE SANTO DOMINGO HAS IT ON TAPE THAT WE WERE CLRED TO 4000 FT NOT 9000 FT. AT THE SAME MOMENT SAN JUAN TOLD US THAT WE WERE 8 MI S OF B520 WHICH HE SAID WE WERE TO BE AT. I TOLD HIM NEGATIVE THAT OUR CLRNC WAS VIA B892-MAZ-G633. HE THEN CLRED US DIRECT DORADO NDB DIRECT SAN JUAN. WHEN WE ARRIVED SAN JUAN; I CALLED THE NUMBER CTR GAVE ME; WHICH WAS CENRAP NUMBER AND THE PERSON I SPOKE WITH TOLD ME THE PROB WAS GOING TO BE INVESTIGATED.