COMPRESSOR STALL AFTER ENG ANTI-ICE PLACED ON. ENG SHUTDOWN. FLC DSNDED TO FL290 WHILE RELAYING THEIR ACTIONS THROUGH ANOTHER ACR BECAUSE THEY WERE NOT IN COM WITH THE ATC SECTOR. ENG WAS RESTARTED AND ACFT RETURNED TO ASSIGNED ALT.

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AT FL390 AIRSPD 250 KTS/MACH .805 ENTERED TOP OF CELL NOT SHOWING ON RADAR. AFTER ENG ANTI-ICE POSITIONED TO ON; L ENG OVERTEMPED TO 742 DEG C; COMPRESSOR STALL EXPERIENCED. THROTTLE MOVED TO IDLE NO HELP. SHUT DOWN L ENG PER PROCS; DSNDED TO FL290. AFTER SUCCESSFUL RELIGHT; CLBED BACK TO FL390. ACR A RELAYED INFO TO DISPATCH. WE WERE NO COM ON VHF/HF ALL FREQS EXCEPT WITH ACR A. BOTH FO WORKED THE PROB AND PERFORMED EXEMPLARY! ENG OPERATED NORMAL AFTER RELIGHT; CONTINUED TO MIA. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: ACFT INDENTED AS B767-200. RPTR STATES THAT THE AREA THEY WERE IN; UPPER LEVEL 304 IS ONE IN WHICH COM IS ALWAYS DIFFICULT BECAUSE SATCOM IS THE ONLY TYPE WHICH PROVIDES COVERAGE. AND; THEY HAD ONLY HF AND VHF; BUT WERE ABLE TO DECLARE AN EMER; DSND AND CLB BACK TO ALT BY RELAYING THE INFO THROUGH ANOTHER ACFT IN THEIR VICINITY. RPTR CAPT BELIEVES WHAT CAUSED THE COMPRESSOR STALL WAS ICE BUILDUP PRIOR TO TURNING ON THE ENG ANTI-ICE. THE REASON THE ANTI-ICE WAS NOT TURNED ON EARLIER WAS BECAUSE THE OUTSIDE TEMP WAS MINUS 28 DEGS. HOWEVER; WHEN THEY FLEW INTO A CELL BUILDUP AND NOTICED MOISTURE; ICE HITTING WINDSCREEN; THEY TURNED ON THE ANTI-ICE. RPTR THINKS BY THE TIME THEY TURNED IT ON; AN APPRECIABLE BUILDUP OF ICE HAD ACCUMULATED ON THE FAN SECTION. AFTER THEY SHUT DOWN THE ENG AND DSNDED; THEY RESTARTED IT AND CONTINUED ON TO DEST. POST FLT INSPECTION FOUND NO DAMAGE; BUT THE ENG HAD A HISTORY OF PREVIOUS OVERTEMPS.