DEP CTLR UNAUTH UNCOORD PENETRATION OF AIRSPACE ACR Y HAD LTSS WITH ACR X. SYS ERROR.

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WHILE AT 4000 FT MSL ON THE F4 LOC OF MDW; JUST OUTSIDE CADON INTXN ACR X TOLD BY APCH TO TURN L 270 DEG. THIS XMISSION WAS VERY QUICKLY GIVEN AND CAME ACROSS VERY GARBLED. THE FO WAS FLYING AND I REQUESTED APCH TO SAY AGAIN. I THOUGHT I HEARD THE CTLR ASK FOR THE TURN BUT WAS NOT SURE SO I LOOKED AT MY FO TO SEE IF HE HEARD THE QUESTION. AS I LOOKED TOWARD MY FO I SAW THE REASON FOR THE VECTOR. ACR Y TYPE ACFT WAS IN A CLBING L TURN AT OUR CLOSE 2 O'CLOCK POS. BY THE TIME I SPOTTED ACR Y IT WAS APPARENT THAT IT WOULD PASS ABOVE AND BEHIND OUR ACFT. HOWEVER; WE WERE WELL INTO THE 270 DEG HDG TURN BY THE TIME THE CTLR REPEATED THE COMMAND TO TURN L 270. WE WERE EVENTUALLY VECTORED BACK ONTO THE LOC TO COMPLETE THE APCH. AN UNIDENTIFIED ACFT ASKED WHAT HAPPENED AND THE CTLR REPLIED THAT THE TURNS WERE FOR A 'NORDO' ACFT. ACR Y ACFT STATED TO APCH THAT 'HE HAD TO CLB TO AVOID THAT ACFT.' UPON LNDG AT MDW I CALLED CHICAGO APCH AND ASKED TO TALK TO THE WATCH SUPVR. I ASKED WHAT HAD OCCURRED. HE STATED THAT ACR Y WAS BEING WORKED BY ANOTHER CTLR AND THAT AN ERROR HAD MOST LIKELY OCCURRED! I SAID THAT I WANTED TO MAKE SURE THAT WE HAD NOT MESSED UP IN ANY WAY AND HE REPLIED THAT WE HAD NOT. HE WANTED TO KNOW WHAT THE WX CONDITIONS WERE AND I TOLD HIM THAT WE WERE IN AND OUT OF CLOUDS BTWN LAYERS. I DID TELL HIM THAT WE WERE IN THE CLR LONG ENOUGH FOR ME AND SOME OF MY 114 PAXS TO SEE ACR Y GO BY. OUR LAST INSTRUCTIONS BEFORE THE CONFLICT WAS TO INTERCEPT THE RWY 4R LOC AND MAINTAIN 4000 FT TILL CADON; CLRED APCH; CONTACT TWR AT ERMIN; MAINTAIN 210 KTS FOR NOW. SUPPLEMENTAL INFO FROM 190866. APCH CAME UP AND SAID TO TURN L TO A 270 DEG HDG. HIS XMISSION WAS GARBLED SO WE ASKED TO CLARIFY AND AT THAT TIME WE SAW ACR Y AT OUR 2 O'CLOCK TAKING EVASIVE ACTION IN A CLBING L TURN THROUGH OUR ALT. WE IMMEDIATELY TURNED L AS WELL. ACR Y PASSED BEHIND US AT ABOUT 1/2 MI RANGE. ACCORDING TO ATC; ACR Y WAS ON DEP FROM MDW AND SOMEHOW WAS ON THE WRONG FREQ AND ALT. ATC BELIEVED A CTLR ERROR WAS MADE. WE CONTINUED OUR APCH AND LANDED AT MDW. THE ACR Y CREW STATED ON THE RADIO THEY SAW US AND HAD TO TAKE EVASIVE ACTION.