ACR EXPERIENCED CONTINUED CALL SIGN MIX-UPS BY CTLRS AT SEVERAL FACILITIES BECAUSE OF SIMILAR CALL SIGNS.

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ATC AT ZDV; ZLA AND LAS APCH KEPT MIXING UP FLT NUMBERS AND OR COMPANY CALL SIGNS BTWN OUR FLT ACR X AAA AND ACR Y BBB. I'VE HAD SIMILAR CALL SIGN INCIDENTS BEFORE BUT NOTHING EVEN CLOSE TO WHAT HAPPENED HERE. I BELIEVE THERE WAS A LOSS OF ATC REQUIRED SEPARATION (NO TCAS EVENT) BECAUSE OF THE REPEATED MISTAKES MADE BY ATC. IT STARTED WITH THE LAST ZDV CTLR WHO CONTINUALLY CALLED US ACR X BBB INSTEAD OF ACR X AAA (ACR Y BBB WAS ALSO ON FREQ). AFTER MULTIPLE CLARIFICATIONS BY US; DSCNT WAS BEGUN INTO LAS; BUT CTLR CONTINUED TO MIX UP FLT NUMBERS. AFTER HDOF TO ZLA WAS MADE THE ERRORS CONTINUED (EITHER FLT NUMBER OR COMPANY CALL SIGN WAS TRANSPOSED). IT GOT TO THE POINT THAT I STOPPED THE DSCNT BECAUSE IT WAS IMPOSSIBLE FOR ME TO TELL WHICH ACFT ATC WAS TALKING TO (EVEN AFTER MULTIPLE CLARIFICATION REQUESTS). ZLA FINALLY GOT THE CALL SIGNS RIGHT THEN GAVE ACR Y AN IMMEDIATE DSCNT AND US AN IMMEDIATE TURN OFF COURSE. THE ACR Y ACFT WAS ACQUIRED VISUALLY BY US AND NEVER CAME CLOSER THAN 5 MI TO US WITH NO TCAS EVENT OCCURRING. UNBELIEVABLY AS IT SOUNDS; ONCE HANDED OFF TO LAS APCH; THE CONFUSION CONTINUED AS CALL SIGNS WERE AGAIN MIXED UP BY ATC FOR THE FIRST COUPLE OF XMISSIONS. WE FINALLY GOT THE APCH CTLR TO CORRECT THE CALL SIGN AND REMAINDER OF FLT WAS UNEVENTFUL.