SCT CTLR CONCERNED WITH NON COORD ACFT IN HIS AIRSPACE THAT CONFLICTED WITH ANOTHER IFR ACFT.

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I HAD 6 ACFT ON FREQ; 3 OF WHICH WERE FOREIGN ACR'S. AS THE MD11 LEVELED AT 12000 FT; THE CONFLICT ALERT ACTIVATED WITH TFC TO THE SW OF HIM CLBING NEBOUND. THE TAG SHOWED HIM ON A CTR TAG CLBING OUT OF 11700 FT; THEN SWITCHING TO AN 'M' TAG. ('M' TAG IDENTS THAT AN APCH SECTOR WITH SCT AIRSPACE TOOK THE HDOF.) NOBODY COORDINATED WITH ME TO ALLOW THE VFR BEECHJET TO CLB THROUGH MY AIRSPACE. BY THE TIME THE CONFLICT ALERT ACTIVATED; THERE WAS NOTHING I COULD DO. SOMEHOW THE ACFT WAS RADAR IDENTED; ALLOWED TO CLB; AND HANDED OFF TO ANOTHER SECTOR WITHOUT APPROVAL; OR TFC ISSUED. I'VE BEEN A CTLR FOR 19 YRS AND IT WAS THE CLOSEST NEAR MIDAIR THAT I'VE EVER SEEN. I DON'T KNOW IF THE VFR BEECHJET HAD THE MD11 IN SIGHT. IF HE DID; THAT PLT SHOULD HAVE BEEN VIOLATED FOR OPERATING TOO CLOSE TO ANOTHER ACFT. THE PLT WAS 'VERY CONCERNED' ABOUT THE SIT; BUT DID NOT CALL THE FACILITY AFTER LNDG. DUE TO THE VOLUME OF TFC; AND MY DESIRE TO SPEAK CLRLY AND A LITTLE SLOWER THAN NORMAL PROBABLY CAUSED ME TO FOCUS ONLY ON THE ACFT ON MY TAGS AND NOT AS MUCH ON THE TAGGED AND UNTAGGED ACFT ON MY SCOPE. THE CONFLICT ALERT DID NOT ACTIVATE AS PROMPTLY AS I WOULD HAVE LIKED. IT ACTIVATED WITHIN 2 MI AND 300 FT BEFORE TARGETS MERGED.