ACR X TCASII TA RA NON ADHERENCE TO ATC CLRNC NON ADHERENCE TO TCASII RA HAD LTSS FROM WDB. EVASIVE ACTION TAKEN. PLTDEV.

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ACR X AT INTERMEDIATE ALT OF FL240; HDG 300 DEGS DIRECT TO COSTA VOR. WE WERE TOLD TO TURN R TO 330 DEGS. SECONDS LATER WE WERE TOLD TO TURN L TO 270 DEGS. WHILE IN THE TURN WE WERE TOLD TO TURN FURTHER L TO 240 DEGS. WE THEN GOT A TA ON THE TCASII SYS. THE TA INDICATED 12 O'CLOCK; 6 NM; 500 FT ABOVE US DSNDING. THE CAPT THEN SAID; 'I HAVE THE ACFT' AND STARTED A DSCNT. WE THEN GOT AN RA INDICATING A CLB WAS NECESSARY. I SAID 'WE NEED TO CLB.' THE CAPT SAID 'I HAVE THE TFC VISUALLY' AS HE REVERSED THE TURN TO THE R WHILE STILL DSNDING. I THEN SAW THE TFC AS IT PASSED ABOUT 500 FT BELOW US OFF OUR L SIDE ABOUT 1000 FT LATERALLY. THE CAPT THEN PASSED CTL OF THE ACFT BACK TO ME AND I STARTED A CLB BACK TO FL240. A DIFFERENT CTLR THEN CLRED US DIRECT COSTA VOR AND GAVE US A FREQ CHANGE. AT NO TIME DID A CTLR MENTION THE TFC OR QUESTION OUR ALTDEV. SUPPLEMENTAL INFO FROM ACN 242204: I DETECTED CONCERN IN THE CTLR'S VOICE AND INSTRUCTED THE FO TO TIGHTEN UP THE TURN. BEFORE PASSING 270 DEGS THE RELIEF PLT NOTICED TFC ON THE TCASII SYS. HE THEN SPOTTED THE TFC VISUALLY AND ADVISED ME TO TURN R TO MISS THE TFC. I SPOTTED THE TFC; TOOK CTL OF THE ACFT AND ENTERED A R TURN. THE OTHER WDB SPOTTED US AND ALSO ENTERED A R BANK AWAY FROM US. WE RECEIVED AN RA ON THE TCASII SYS DURING THIS TURN. HOWEVER; I WAS VISUALLY AVOIDING ACR Y AND DID NOT CLB SINCE I WOULD HAVE TO LOSE SIGHT OF ACR Y AND THEN TRANSITION BACK TO INSIDE THE COCKPIT.