EMB120 ACFT ENTERING TFC PATTERN HAD A TCASI ALERT CAUSED BY A GRUMMAN TIGER ACFT NOT FOLLOWING PATTERN ENTRY INSTRUCTIONS ISSUED BY TWR CTLR.

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FROM TPA TO EYW. CLRED FOR VISUAL APCH BY NAVY KEY WEST APCH ABOUT 10 MI N OF VORTAC TO RWY 9. CHANGED FREQ TO TWR ABOUT 7 MI N OF VORTAC. INSTRUCTED TO RPT ABEAM VORTAC ON L BASE. APCHING THE VORTAC TWR CTLR WAS BUSY SCOLDING ANOTHER ACFT FOR NOT FOLLOWING HER INSTRUCTIONS. THE ACFT WAS SUPPOSED TO BE SE OF THE ARPT AND TOLD TO ENTER R DOWNWIND TO RWY 9. WHILE WAITING FOR AN OPENING TO XMIT OUR POS RPT AS REQUESTED; WE RECEIVED A TA FROM OUR TCASI SAYING 'TFC; TFC!' THE SCREEN SHOWED AN INTRUDER GREATER THAN 1/2 MI AND 100 FT BELOW US. WE IMMEDIATELY INCREASED OUR VISUAL SCAN AND ASKED THE TWR CTLR FOR ANY ACFT IN OUR VICINITY. SHE SAID THAT A GRUMMAN WAS SUPPOSED TO BE ON OTHER SIDE OF ARPT BUT SHE COULD NOT SPOT HIM VISUALLY. THE GRUMMAN THEN RPTED US IN SIGHT AND SAID THAT HE'D FOLLOW US TO THE RWY. WE DID NOT SEE THE AIRPLANE UNTIL WE CLRED THE RWY AFTER LNDG. WE WERE COMPLETING THE BEFORE LNDG CHKLIST WHEN TCASI ALERTED US DURING THIS CRITICAL PHASE OF FLT AND PERHAPS WERE NOT AS AWARE OF THIS OTHER ACFT'S POS AS WE SHOULD HAVE BEEN. TWR CTLED RADAR COULD HAVE PREVENTED THIS NEAR MISS AND INCIDENTS OF THIS NATURE IN THE FUTURE.