A B737-800 FLT CREW RPTED THAT; DURING DEP FROM TUL; THEY EXPERIENCED AN NMAC WITH ANOTHER ACFT THAT HAD NOT BEEN POINTED OUT BY TUL TRACON.

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DURING CLBOUT FROM TUL (RWY 18L) 178 DEGS RWY HDG TO 15000 FT; TULSA DEP ISSUED US A TFC ALERT AT OUR 1 - 1:30 O'CLOCK POS; TYPE AND ALT UNKNOWN. WE WERE TOLD THAT 'IF THE TFC WAS NOT IN SIGHT; SUGGEST YOU LEVEL YOUR ACFT NOW.' HAND FLYING; USING THE HUD; I PUT THE FLT PATH VECTOR ON THE HORIZON TO IMMEDIATELY LEVEL US OFF WHILE WE TRIED TO VISUALLY ACQUIRE THE TFC. AT THAT POINT OUR TCASII GAVE US A TA; WITH THE AURAL WARNING; 'TFC; TFC.' IT THEN CHANGED TO RED WITH AN RA TO 'CLB; CLB' WITH AN ADDITIONAL WARNING TO 'INCREASE CLB.' HAND FLYING; I WAS ABLE TO FOLLOW THE SYMBOLOGY IN THE HUD WITHOUT ANY DELAY. I THEN HEARD; 'MONITOR VERT SPD.' I HELD WHAT I HAD UNTIL THE 'CLR OF CONFLICT' WAS HEARD. THE INITIAL TA SHOWED THE TARGET AT -300 FT AND CLOSING. THE RA IN RED SHOWED THE TARGET AT -100 FT PASSING UNDER THE ACFT SYMBOL ON OUR DISPLAY. AFTER THE CONFLICT WAS RESOLVED; I RESUMED THE NORMAL CLB PROFILE TO ALT. ATC WAS NOT SURE WHY THE TARGET ACFT SHOWED UP ALL OF A SUDDEN. I WAS TOLD THAT HE THINKS HE JUST TURNED HIS XPONDER ON WHEN WE RECEIVED THE ALERTS. WE NEVER SAW THE ACFT. TULSA DEP DID RAISE HIM ON THE FREQ AND I HEARD THE TARGET ACFT SAY 'WE HAVE THE AIRLINER IN SIGHT AND WILL MAINTAIN VISUAL SEPARATION.' AT THAT POINT HE WAS NO LONGER A THREAT AND WE PROCEEDED TO DFW ON COURSE. UPON ARR AT THE GATE; I CALLED THE DISPATCHER TO FILE THE RPT AND COMPANY WANTED TO HAVE A CONFERENCE CALL WITH TULSA APCH/DEP. A SUPVR CAME ON THE PHONE AND TOLD US THAT; AS PER THEIR PROCS; THE CALL WILL BE RECORDED. WE DISCUSSED THE EVENTS THAT HAPPENED ON CLBOUT AND THEN TOLD HIM THAT I HAD NO CHOICE BUT TO FOLLOW THE TCASII RA. I BELIEVE THAT HAD I NOT CLBED; WE WOULD HAVE PASSED EVEN CLOSER; OR POSSIBLY COLLIDED. THE TCASII WORKED AND PERFORMED FLAWLESSLY. SUPVR SAID THE TAPES HAVE BEEN PULLED AND WILL BE REVIEWED. SUPPLEMENTAL INFO FROM ACN 624739: CLBING OUT OF TUL; THROUGH ABOUT 8000 FT CLRED TO 15000 FT; CAPT FLYING. TALKING TO TUL DEP. NOTICED A TARGET POP UP ON THE TCASII; WHICH I HAD ON THE 10 MI RANGE. IT WAS VERY CLOSE IN; READING +06. I SAID 'HEY; WHAT'S THIS GUY?' JUST AS ATC CALLED 'TFC ALERT.' BY THIS TIME I HAD PUT THE RANGE ON 5 MI; ONLY TO SEE THE TARGET SHOWING +02; SO CLOSE TO OUR ACFT SYMBOL; THE BEARING WAS INDISCERNIBLE. CTLR SAID; 'IF YOU DON'T SEE HIM; SUGGEST YOU LEVEL IMMEDIATELY.' WE WERE DOING ABOUT 3000 FPM AT THE TIME; AND THE CAPT; WHO WAS HAND FLYING; IMMEDIATELY BEGAN TO LEVEL. ABOUT THIS TIME; THE TCASII ISSUED AN RA SAYING; 'CLB; CLB' WITH MOST OF THE VSI RED; EXCEPT FOR THE PORTION ABOVE ABOUT 2800 FPM. BECAUSE WE HAD JUST BEEN TOLD TO LEVEL IMMEDIATELY AND WERE IN THE PROCESS OF DOING SO; IT TOOK US A FEW SECONDS TO PROCESS THIS INFO AND MAKE THE TRANSITION. THE SEEMINGLY CONTRADICTING INFO WAS DUE TO THE FACT THAT WE HAD PASSED THROUGH THE ALT OF THE OTHER ACFT DURING OUR ATTEMPT TO LEVEL; AND NOW HE WAS BELOW US; ALTHOUGH WE COULD NOT SEE THAT BECAUSE THE TARGET WAS SUPERIMPOSED ON OUR ACFT SYMBOL ON THE TCASII SCREEN. ABOUT THIS TIME; THE TCASII ISSUED A SECONDARY WARNING; SOMETHING LIKE 'INCREASE CLB RATE;' ALTHOUGH I DO NOT RECALL THE EXACT VERBIAGE. AFTER WE WERE CLR OF THE CONFLICT; WE QUERIED ATC; WHO INDICATED THAT THIS TFC POPPED UP ON HIS SCREEN THE SAME TIME HE POPPED UP ON OURS; AND THAT HE WAS NOT IN CONTACT WITH HIM AT ALL. HE ALSO INDICATED THAT THE ACFT WAS 'ON THE EDGE' OF BEING ILLEGAL IN THE AIRSPACE; AND HE WOULD 'LOOK INTO IT.' WE NEVER DID SEE THIS ACFT; BUT WE CAME WITHIN A FEW HUNDRED FT LATERALLY AND WITHIN ZERO FT VERTLY. THIS WAS WAY TOO CLOSE.