WHILE ON APCH TO THE MEM W SIDE; A CL65 CREW NOTICED THE LCL FREQ WAS SPLIT LEAVING THEM ON THE E SIDE FREQ.

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I WAS FLYING FROM DFW TO MEM. WE GOT CLRNC FOR ILS RWY 36L IN IFR CONDITIONS. AT THE TIME MEM WAS BEING CTLED BY 1 LCL CTLR FOR THE ENTIRE ARPT. ONCE WE RECEIVED LNDG CLRNC AND ESTABLISHED INSIDE THE FAF; MEM OPENED A SECOND LCL CTLR FOR THE W SIDE OF THE ARPT (RWY 36L). WE NEVER RECEIVED NOTIFICATION TO SWITCH FREQ SINCE WE WERE WITH THE E SIDE CTLR. TWR TOLD US BEFORE FAF TO EXPECT 1 DEP PRIOR TO OUR ARR. WHEN WE BROKE OUT OF THE CLOUDS ON ILS RWY 36L; WE SAW ANOTHER AIRPLANE ON THE RWY (AN A320). WE NEVER HEARD HIS TKOF CLRNC SINCE THE TWR HAD SPLIT THE FREQS AND WE WERE ON THE E SIDE FREQ ON THE W SIDE RWY. UPON REACHING 200 FT AGL AND THE OTHER AIRPLANE STILL ON THE RWY; WE EXECUTED MISSED APCH. THE CTLR WAS STARTLED SINCE HE WAS CTLING THE E SIDE OF THE ARPT. WE DID NOT RECEIVE IMMEDIATE INSTRUCTIONS SO I BEGAN TO EXECUTE THE PUBLISHED MISSED APCH PER ILS RWY 36L INST APCH PROC. THE CAPT (WHO WAS THE PNF) TOLD THE TWR THAT WAS 'ON THE GO' INSTEAD OF USING STANDARD ATC PHRASEOLOGY; WHICH FURTHER CONFUSED THE CTLR. WE FINALLY RECEIVED INSTRUCTIONS FROM ATC TO CLB TO 2000 FT AND TURN 190 DEG HDG. CAME BACK AROUND AND DID ILS RWY 36L WITHOUT INCIDENT FOLLOWED BY UNEVENTFUL LNDG.