ACR LGT CLOSE ENCOUNTER WITH GOV LTT NEAR THE FINAL APCH FIX FOR RWY 28R AT SFO.

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WE HAD BEEN CLRED FOR THE ILS 28R AT SAN FRANCISCO AND HAD INTERCEPTED LOC AND G/S. DSNDING THROUGH ABOUT 4000' ON G/S; APCH ASKED IF WE SAW A GOV LTT AT 12 O'CLOCK AT 1500'. WE SAW HIM AS RPTED; HE WAS E OF BRIJJ AND SBND. WE LOST SIGHT OF HIM AS HE PASSED UNDER TH NOSE; WE THEN ENTERED A CLOUD LAYER APCHING BRIJJ OUTER MARKER AT ABOUT 2000'. WHEN WE BROKE OUT BELOW THE CLOUD LAYER INSIDE BRIJJ AT ABOUT 1600'; THE LTT WAS ON OUR LEFT WING AT THE SAME ALT AND NO MORE THAN 100 OR 200' LATERALLY; PARALLELING OUR COURSE. WE HAD SWITCHED TO TWR FREQ AT BRIJJ; AND TWR ASKED IF WE STILL SAW THE GOV LTT. WE CONTINUED DOWN THE G/S; AND HE MAINTAINED LEVEL FLT; THEN MADE A RIGHT TURN; PASSING OVER US BY A FEW HUNDRED FEET. IF THE GOV LTT WAS UNDER IFR CONTROL; HE SHOULD NOT HAVE BEEN SO CLOSE TO US. IF HE WAS MANEUVERING VFR; HE DID NOT MAINTAIN VFR CLOUD CLRNC CRITERIA. NEITHER APCH CTL NOR TWR INDICATED THAT THE GOV LTT WOULD BE TURNING TOWARD THE ARPT; OR THAT HE WOULD FOLLOW THE LOC; OR THAT WE SHOULD KEEP HIM IN SIGHT; AND NO FOLLOWUP POS INFO WAS OFFERED TO US. ONCE HE PASSED UNDER OUR NOSE SBND WE THOUGHT WE WERE CLR.