IFR C172 VECTORED TO ILS BY GRB CTLR EXPERIENCED NMAC WITH VFR ACFT.

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I WAS ON AN IFR FLT PLAN IN VMC FROM 57C TO ATW. I WAS HANDED OFF FROM ZAU TO GRB APCH. SHORTLY AFTER THE HDOF; GRB ASKED ME IF I HAD TFC AT MY 3 O'CLOCK POS; IN REAL CLOSE. I CHKED AND COULDN'T SEE ANYTHING. THEN GRB SAID THE TARGET DISAPPEARED AND THEY SUSPECTED A 'SPLIT BEACON.' I WAS ON A 360 DEG VECTOR UP THE E SIDE OF LAKE WINNEBAGO; BEING VECTORED FOR THE ILS RWY 29 AT ATW. THE INTERCEPT TAKES PLACE ON THE N SIDE OR THE LAKE. GRB TURNED ME IN ON THE INTERCEPT AND CLRED ME FOR THE APCH. I WAS AT 3000 FT. SHORTLY AFTER MAKING THE TURN INBOUND; THEY TOLD ME TO CALL ATW TWR. AT THE SAME TIME; I SPOTTED THE OTHER PLANE DIRECTLY ABOVE ME AND SLIGHTLY AHEAD. WE HAD 300-500 FT OF VERT SEPARATION. I RPTED IT TO GRB APCH; AND THEY SAID IT MUST NOT HAVE BEEN A SPLIT BEACON AFTER ALL. OUR 2 TARGETS WERE MERGED THE ENTIRE TIME. GRB SAID THEY WERE NOT TALKING TO HIM. WHEN I SWITCHED TO ATW TWR; I RPTED IT. THEY SAID THEY WERE TALKING TO HIM AND HE WAS SIGHT-SEEING AROUND THE SHORE OF THE LAKE. BECAUSE I WAS THE LOWER PLANE IN A HIGH WING; AND HE WAS HIGHER IN A LOW WING; WE NEVER SAW EACH OTHER.