MMMX LCL CTLR ISSUES GAR TO A B737 ON ONE HALF MI FINAL WHEN COMPANY MD80 ENTERS RWY 5 CLR AREA AFTER ACKNOWLEDGING COMPANY'S CALL SIGN CLRNC.

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OUR FLT; ACR X; WAS DELAYED ON TXWY DUE TO OVERHEAD AND DEP TSTMS/RAINSHOWERS. WE WERE ON TXWY A SHORT OF RWY 5R. FOR 1 HR; WE WERE THE ONLY ACR X ON THE RADIO. OVERHEAD STORM PASSED; BUT DEP DIRECTION ENGULFED WITH TSTMS. AS TSTMS MOVED AND LESSENED; SOME MEXICAN PLANES BEGAN TO DEPART AND ARRIVE. WE ASKED MEX TWR TO GIVE US A PIREP FROM DEPARTING MEXICAN ACFT; SINCE WE DIDN'T SPEAK SPANISH. AFTER 15 MINS; TWR GAVE US A RPT THAT DEPARTING ACFT (RWY 5) WERE MAKING N TURNOUTS AT 4-5 DME TO CLR TSTMS. THIS WOULD WORK WITH OUR ENG FAILURE PROC; SO WE REQUESTED CLRNC FOR TKOF. MEX TWR REPLIED; 'OK ACR X FLT ABCD; STAND BY.' THEN WE HEARD 'ACR X FLT AACD; CLRED RWY 5R' AND BOTH MYSELF AND THE FO MISTOOK THAT AS OUR CALL SIGN AND CLRED TO TAXI ONTO RWY 5R. THE FO RPTED BACK 'ACR X FLT ABCD; POS RWY 5R.' MEX TWR SAID 'ROGER;.' WE TURNED OFF TXWY A ONTO TXWY AA; WHICH IS THE 700 FT LEAD-IN TO RWY 5. AS WE ENTERED TXWY AA; I COULD SEE A B737 ON A 2 MI FINAL AND SIMULTANEOUSLY HEARD TWR SAY 'ACR X FLT ABCD; YOU WERE NOT CLRED ONTO RWY 5R.' I WAS ALREADY MAKING A QUICK TURN L OFF OF TXWY AA ONTO TXWY A1 BEHIND ANOTHER ACR JET HOLDING SHORT OF RWY 5L. I WAS ALREADY TURNED WHEN TWR SAID 'ACR X FLT ABCD; TAXI STRAIGHT AHEAD TXWY A.' I COULD NOT SWING BACK TO FOLLOW TWR'S DIRECTIONS QUICK AND SAFE ENOUGH IN OUR PERSPECTIVE. WHEN WE PULLED UP SAFELY BEHIND THE OTHER ACR TO A STOP; I COULD STILL SEE THE B737 ON APCH AT 1/2+ MI OUT MY L WIND. THEN TWR COMMANDED 'ACR X FLT AACD; GO AROUND.' THIS REALLY SURPRISED ME DUE TO THE WX AT DEP AND ACR X FLT AACD HAD AMPLE ROOM ON APCH. WE WERE WELL AWAY FROM TXWY AA/RWY 5R. CAUSES NOTED: 1) LANGUAGE BARRIER IN MONITORING DEPARTING AND ARRIVING ACFT COMS; ESPECIALLY IN ADVERSE WX. 2) THE LONG SIT AND BEING THE ONLY ACR X ACFT HEARD ON THE RADIO. 3) THE LIKE SOUNDING ACR X FLT NUMBERS -- #ABCD AND #AACD. SUPPLEMENTAL INFO FROM ACN 564686: I WAS FO AND WORKING RADIOS. I RADIOED TWR AND RPTED THAT WE WERE READY AT WHICH POINT TWR REPLIED; 'ACR X FLT ABCD STANDBY;' WHICH I ACKNOWLEDGED THAT. NEXT; I HEARD 'ACR X FLT AACD CLRED RWY 5R' TO WHICH POINT I MADE A WRONG ASSUMPTION AND READ BACK TO MEXICO CITY TWR OUR CALL SIGN. I FEEL THE LANGUAGE BARRIER CONTRIBUTED TO THE CONFUSION AS WELL AS RUNNING LATE AND SIMILAR SOUNDING CALL SIGNS.