AN ACR WDB TOOK EVASIVE ACTION IN RESPONSE TO TCASII BECAUSE OF AN SMA THAT WAS VFR IN IMC.

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DSNDING TO 1700 FT INTO FLL; WE HAD BEEN CLRED DIRECT TO PIONN. MIAMI APCH RPTED VFR TFC AT 11-12 O'CLOCK; 4 MI; NWBOUND AT 3000 FT. WE WERE SEBOUND. THE TFC APPEARED AS A SOLID WHITE DIAMOND AT ABOUT 4 MI AT 3000 FT. WE ASSUMED HE WAS VFR BELOW THE CLOUDS. I TURNED ON ALL THE LIGHTS WE HAD. THE TARGET'S ALT CHANGED TO 3100 FT AND WENT RAPIDLY TO YELLOW AND THEN RED AS HE CLBED. WE WERE REDUCING AIRSPD FROM 250 TO 200 KTS THINKING THAT WOULD HELP KEEP US ABOVE HIM; ESPECIALLY IF HE WAS IN FACT VFR. (FLL WX WAS RPTED AT 2500 FT BROKEN AT THAT TIME). WHEN THE TCASII WENT RED; I TURNED ABOUT 50 DEGS TO THE R -- 30 DEG BANK -- AND LEVELED OFF. THE TCASII IVSI RAPIDLY CHANGED TO A NEARLY FULL RED DISPLAY COMMANDING A CLB IN EXCESS OF THE CAPABILITY OF THE ACFT AND ENGS TO RESPOND. WE DID CLB 300 OR 400 FT AS THE TARGET ON THE TCASII PASSED OFF TO OUR L SIDE IN CLOSE PROX (WELL LESS THAN A MI). WE NEVER SAW THE TFC. HE WAS AN SMA HAVING JUST DEPARTED N PERRY. HE TOLD APCH HE WAS VFR THE WHOLE TIME; WHICH WAS NOT POSSIBLE. HE ALSO STATED HE NEVER SAW US -- WHICH IS HARD TO BELIEVE COULD HAVE HAPPENED IF HE PASSED WITHIN A MILE -- VFR -- WITH ALL OF OUR LIGHTS ON AT NIGHT. THERE IS NO DOUBT IN OUR MINDS THAT HE CLBED; IMC; ON A VFR CLRNC; THROUGH OUR ALT NEARLY CAUSING A MIDAIR. HE WAS TALKING TO ZMA (WHOSE RADAR WAS OUT); WE WERE ON FLL APCH. AFTER WE PASSED HIM; WE TURNED TOWARDS FLL AND INTERCEPTED LOC BREAKING OUT OF CLOUDS AT 2500 FT -- CLR BELOW.