B737-700 CREW HAD LESS THAN LEGAL SPACING IN ZKC CLASS A AIRSPACE.

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POSSIBLE LOSS OF SEPARATION. LEVEL CRUISE AT FL370 PROCEEDING DIRECT TO BKW. APPROX OVER TUL; WE HEARD CTLR ISSUE A TURN TO A COMPANY ACFT. WE WERE TALKING AND RADIO TFC WAS VERY HVY. WE MISSED THE CALL SIGN AND WAITED A MOMENT FOR CTLR TO REISSUE CLRNC. WHEN CTLR REISSUED CLRNC HE WAS GARBLED/MUFFLED AND IT SOUNDED TO ME LIKE HE WAS CALLING 'ACR XYZ'.' WHEN 'ACR XYZ' DID NOT RESPOND; I QUERIED THE CTLR AS TO WHO THE CLRNC WAS INTENDED FOR. AT THIS POINT; THE CTLR SOUNDED TENSE AND TOLD US TO TURN IMMEDIATELY L TO 350 DEGS FOR TFC. THIS REPRESENTED ALMOST A 90 DEG TURN FROM OUR TRACK OF 075 DEGS AND STRUCK US BOTH AS UNUSUAL. THE CTLR CALLED OUT OPPOSITE DIRECTION TFC CLBING AND I SPOTTED IT AT THE SAME TIME WE RECEIVED THE TA. TCASII INDICATED THE CONFLICT ACFT TO BE +1500 FT AND THE ACFT RPTED CLBING THROUGH FL385 CONFIRMING THIS. (WHEN ISSUED THE CLRNC TO TURN; THE CAPT DID SO WITHOUT DELAY.) I DO NOT BELIEVE A CONFLICT WAS IMMINENT; BUT SEPARATION MAY HAVE BEEN COMPROMISED. OUR TURN WAS DELAYED BY POOR COM; BOTH THE CAPT AND I HEARD WHAT WE THOUGHT WAS ANOTHER CALL SIGN WITH OUR TURN CLRNC. CTR RADIO WAS NOT CLR -- STATIC AND SLIGHTLY GARBLED. I DO NOT NOW IF HE USED THE WRONG CALL SIGN OR IF THE POOR XMISSION QUALITY WAS RESPONSIBLE. ANOTHER CTLR TOOK OVER ON THE SAME FREQ AND WAS CLR AND READABLE. THE SIT MAY HAVE BEEN AVOIDED IF THE CAPT AND I HAD NOT BEEN IN CONVERSATION INITIALLY; BUT I DON'T THINK WE MISSED OUR CALL SIGN. ALSO; I SUSPECT THE OPPOSITE DIRECTION TFC WAS NOT ACHIEVING THE CLB PERFORMANCE THE CTLR HAD ANTICIPATED. (BUT THIS IS JUST SPECULATION ON MY PART.)