ZJX CONTROLLER MIXES CALL SIGNS AND DESCENDS A NORTHBOUND B737 INTO THE PATH OF A SOUTHBOUND B747 AT FL380.

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APPROX 120 MI S OF DIW AT FL390; ZJX BROADCASTS 'COMPANY FLT NUMBER DSND TO FL380 FOR OVERTAKING TFC BEHIND YOU.' WE RESPOND 'COMPANY FLT NUMBER DSNDING TO 380.' AT APPROX FL384; WE GET A TCAS 'TFC; TFC' ALERT AND SEE A B747 GO BY OPPOSITE DIRECTION AT FL380 BUT SUFFICIENTLY AWAY NOT TO CREATE A TCAS RA. ABOUT 15 SECONDS LATER JACKSONVILLE BROADCASTS 'COMPANY FLT NUMBER PLEASE CONFIRM THE ALT YOU ARE AT.' WE ANSWER THAT WE ARE AT THE ASSIGNED ALT OF FL380. JACKSONVILLE ANSWERS THAT THAT CLRNC WAS FOR COMPANY NUMBER AND WE ARE TO TURN 20 DEGS TO THE R AND CLB BACK TO FL390. WE COMPLY WHILE JACKSONVILLE GIVES THE 'DSND TO FL380' CLRNC TO COMPANY FLT NUMBER FOR OVERTAKING TFC. COMPANY FLT NUMBER RESPONDED AS IF THEY RECEIVED THE CLRNC FOR THE FIRST TIME. THE TURN FOR US WAS MOOT BECAUSE THE B747 HAD ALREADY GONE BY; BUT WE COMPLIED. AFTER JACKSONVILLE SORTED IT OUT AND CLRED US BACK DIRECT TO SNOW HILL VOR; HE APOLOGIZED FOR MIXING THE CALL SIGNS UP AND SAID THERE WAS NO CONFLICT. CONTRIBUTING FACTOR: 2 SIMILAR CALL SIGNS IN THE SAME AIRSPACE. NO OTHER PROBS WERE ENCOUNTERED FOR THE REMAINDER OF THE FLT. PLEASE CONSIDER LOOKING AT SIMILAR CALL SIGNS AND TRY TO DE-CONFLICT THEM GEOGRAPHICALLY. THIS IS NOT THE FIRST TIME I HAVE WRITTEN ABOUT THIS AS A PROB.