LTT MAKES APCH WRONG ARPT. TWR CTLR SEES PROBLEM AND HAS THEM GAR ON HDOF TO TRACON.

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APPROX 15 TO 20 MI OUT OF GTF; THE CAPT (PNF) OBTAINED THE ATIS THEY WERE LNDG RWY 03. AT APPROX 20 MI OUT I CALLED THE RWY IN SIGHT WHICH I; AT THE TIME; THOUGHT WAS GTF; UNKNOWN TO ME IT WAS MALMSTROM AIR FORCE BASE. IN THE ACFT ON THE COPLT'S SIDE AT LOW ALT; YOU CAN'T SEE WHAT'S ON THE CAPT'S SIDE. WE WERE HEADED RIGHT AT MALMSTROM IT LOOKED TO BE THE DISTANCE AS WHAT WAS ON THE DME AND I COULD NOT SEE GTF AND OUR NAV INDICATED THAT IT WAS THE CORRECT ARPT. WE INFORMED APCH THAT WE HAD THE ARPT IN SIGHT. HE; IN TURN; ISSUED TFC THAT WE WERE TO FOLLOW; AT OUR 2 O'CLOCK POS. IT WAS A SMA. THE CTLR STATED THAT THE SMA WOULD BE MAKING A CLOSE IN APCH; AND THAT WE WOULD BE SEQUENCED IN BEHIND HIM. WE NEVER CALLED THE TFC IN SIGHT; BUT I THINK THE WORDING OF THE CTLR'S STATEMENT SERVED TO INCREASE OUR PERCEPTION THAT WE WERE LOOKING AT GTF AND NOT MALMSTROM. A SLOWER ACFT (SMA) BEING POINTED OUT AS TFC WE WERE TO FOLLOW; CALLED OUT TO US IN OUR 2 O'CLOCK POS (MALMSTROM WAS IN OUR 2 O'CLOCK POS ALSO) LED US TO BELIEVE HE WAS 'INSIDE' US. ADDITIONALLY; MOMENTS LATER; WE WERE GIVEN A R TURN TO A HDG OF 310 DEGS; WHICH FURTHER POINTED US DIRECTLY AT THE APCH END OF RWY 03 AT MALMSTROM; AND FURTHER AWAY FROM GTF. THEN; WE WERE CLRED FOR THE VISUAL. I CAN'T RECALL IF THE APCH CTLR STATED FOR A SPECIFIC RWY OR NOT. AT THAT POINT; WE BEGAN A SWEEPING OR ARCING R TURN TO PUT US ON FINAL FOR RWY 03 AT WHAT WE THOUGHT WAS GTF ARPT. WE WERE ALMOST COMPLETELY LINED UP ON FINAL FOR RWY 03 AT MALMSTROM; WHEN THE CTLR SWITCHED US TO TWR FREQ. AGAIN; I CANNOT REMEMBER IF HE SAID 'CONTACT GREAT FALLS TWR' OR 'CONTACT TWR.' WHEN WE CONTACTED GTF TWR; SHE (THE TWR CTLR) QUICKLY SURMISED THAT WE WERE AT MALMSTROM; ISSUED US A GAR; AND SWITCHED US BACK TO APCH. APCH HAD US MAKE AN IMMEDIATE L TURN FOR TFC (WHICH WE LATER LEARNED WAS NO FACTOR) VECTORED US TO RWY 21 AT GTF AND WE LANDED WITHOUT INCIDENT. UNFORTUNATELY FOR US; WE HAD A VIP ON BOARD. BUT; THIS FACT WAS KNOWN; APPARENTLY; TO THE CTLRS. THAT; COMBINED WITH OUR (POSSIBLY) UNFAMILIAR CALL SIGN 'XXX;' WE THINK LED THE APCH CTLR TO BELIEVE WE WERE LNDG MALMSTROM; NOT GTF.