FLC OF AN MLG STOPPED CLB AND DSNDED IN RESPONSE TO A TCASII RA.

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IN LEVEL CRUISE AT FL290; ZOA ASKED FOR OUR HDG. COPLT ANSWERED 114 DEGS. ZOA SAID 'TURN R 15 DEGS AND CLB TO FL330 FOR TFC.' COPLT READ BACK CLRNC AND WE BEGAN TO COMPLY. AS WE CLBED THROUGH FL297; I NOTICED TCASII DISPLAY OF TFC AT 12 O'CLOCK AT FL310. I ASKED COPLT TO CONFIRM OUR CLRNC; BUT ZOA WAS BUSY REISSUING OUR CLRNC TO ANOTHER FLT WITH CALL SIGN VERY CLOSE TO OURS. AT FL298 WE GOT A TCASII ADVISORY; SO I DISCONNECTED AUTOPLT AND LEVELED OFF. AT ABOUT FL299; TCASII ISSUED A DSND TA. ABOUT THIS TIME; ZOA CLRED US TO MAINTAIN PRESENT ALT AND TURN L 070 DEGS. I COMPLIED WITH THE TURN AND FOLLOWED THE TCASII DSCNT ADVISORY UNTIL IT GAVE A 'CLR OF CONFLICT' ABOUT FL297. I BELIEVE THE MAJOR CONTRIBUTING FACTOR TO THIS INCIDENT WAS THE SIMILAR CALL SIGNS OF THE 2 FLTS. I THINK THAT ZOA GAVE US THE CLRNC FOR THE TURN AND CLB; THINKING HE WAS GIVING IT TO THE OTHER FLT. PLEASE NOTE THAT THE COPLT USED OUR FULL CALL SIGN IN ALL XMISSIONS AND BOTH OF US HEARD OUR CALL SIGN IN ZOA'S XMISSIONS.