AFTER RECEIVING DSCNT CLRNC; CTLR URGENTLY TOLD FLC TO CLB AND TURN. FLC RECEIVED TCASII WHICH CONFLICTED WITH CTLR'S INSTRUCTIONS. CTLR WAS RELIEVED BY A SUPVR. FLC CONTACTED ATC AFTER LNDG FOR REVIEW.

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WE WERE AT FL290. RIDE WAS GETTING CHOPPY. REQUESTED FL270. WE WERE CLRED TO FL270. AT APPROX FL285; CTLR TOLD US TO MAINTAIN FL290 AND TURN R 090 DEGS. SENSING URGENCY IN HIS VOICE; I ENGAGED ALT HOLD; SELECTED 090 DEGS AND STARTED CLB BACK TO FL290. THIS ALL TOOK ABOUT 3 SECONDS. FO WAS FLYING THE ACFT AND CONTINUED COMPLYING WITH ATC CLRNC. APCHING FL290; WE GOT A TCASII TA; AND I REMEMBER SEEING +400 ON THE TARGET. ABOUT THIS TIME TCASII CHANGED TO RA AND COMMANDED 'DSND.' FO STARTED TO PUSH OVER; DECK ANGLE WAS APPROX 15 DEGS. I INFORMED CTLR WE HAD A TCASII RA COMMANDING A DSCNT. HE VERY URGENTLY REPEATED FOR US TO CLB. I GRABBED THE YOKE AND PULLED TO A LIGHT BUFFET. AT THIS POINT THE TARGET MERGED WITHOUT ACFT SYMBOL -- OUR ACFT SYMBOL -- I THEN SAW TARGET PASS BEHIND AT -700. I ASKED CTLR TO EXPLAIN WHEN HE GOT A CHANCE. HE APPARENTLY WAS RELIEVED AND THE NEXT CTLR COULD GIVE US NO FURTHER INFO. I CONTACTED MR X AT ZDC. HE IS SUPVR SATCS AND IS CONDUCTING AN INVESTIGATION. NO INJURIES OCCURRED TO THE PAX OR CREW. PAX BELIEVED THAT INCIDENT WAS A TURB ENCOUNTER.