MD80 ON FINAL DFW RWY 36L RPTED A TCASII EVENT; ATC UNABLE TO IDENT ACFT.

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ON FINAL TO DFW; ILS RWY 36L; AUTOPLT COUPLED. AT 11 DME; WE RECEIVED A TA; FOLLOWED WITHIN A FEW SECONDS BY AN RA. WE WERE CAPTURED AT 3000 FT; PRIOR TO GS CAPTURE. THE FIRST RA INSTRUCTED A 1500 FT DSCNT; AND THE RED TARGET ACFT WITHIN 200 FT OF OUR ALT. ALMOST IMMEDIATELY; THE RA WAS CHANGED TO A 2000 FT + CLB. BY THE TIME THE FO HAD DISCONNECTED THE AUTOPLT; IT HAD CHANGED TO A CLB. WE BOTH INSTANTLY SAID CLB. THIS ALL WAS WITHIN JUST 3-5 SECONDS. HE IMMEDIATELY WENT INTO A CLB; AND BY 3400 FT WE HAD AN ALL CLR. I HAD JUST CHKED IN WITH TWR; AND WE RECEIVED THE TA FROM OUR ACFT AT ALMOST THE SAME TIME TWR SAID THEY HAD JUST RECEIVED A PRIMARY TARGET NEAR US. BY THE TIME THE TWR CTLR STOPPED WITH THE ALERT; WE WERE RESPONDING TO THE RA. I TOLD TWR WE WERE RESPONDING TO AN RA AND WERE CLBING TO 4000 FT. THE FO HAD LEVELED OFF AND WAS RETURNING TO 3000 FT. TWR ASKED IF WE WANTED TO CONTINUE THE APCH. I TOLD THEM NO; AND WE'D LIKE TO DO A 360 DEG AND TRY THE APCH AGAIN. WE WERE GIVEN INSTRUCTIONS TO CONTINUE TO FLY THE LOC AND CLB TO 4000 FT. WE DID; AND RETURNED FOR AN UNEVENTFUL APCH AND FULL STOP. AFTER WE GOT IN THE TERMINAL; WE WENT TO OPS TO TALK WITH A CHIEF. I ALSO CALLED THE W TWR; TALKED TO THE SUPVR; AND ASKED IF HE HAD SEEN THE INCIDENT; WHICH HE HAD. INITIALLY; HE THOUGHT IT WAS A VFR ACFT TRYING TO SNEAK INTO GRAND PRAIRIE ARPT. HE CALLED ME LATER AND SAID THAT APCH THOUGHT IT MIGHT HAVE BEEN A BAD HIT (POOR WORDING) FROM THE MODE C. IN ANY CASE; WE WERE IFR AND DIDN'T SEE ANY ACFT; AND DIDN'T NOTICE ANY HOLES IN THE OVCST.