B737 DEP FROM FLL EXPERIENCED TCASII RA WHILE ATC IS CONFUSED WITH ACFT LOCATION.

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FLL CLRED US FOR TKOF AND FLY RWY HDG (ACTIVE RWY 27R) HDG 276 DEGS. PER CLRNCS INSTRUCTION; WE LEVELED AT 3000 FT MSL. WHILE LISTENING FOR TWR TO SWITCH US TO DEP; I HEARD TWR CLR US ONTO THE RWY; POS AND HOLD. I DID NOT RESPOND BECAUSE I THOUGHT HE WAS MISTAKEN SINCE WE WERE ALREADY AT 3000 FT MSL. HE THEN SAID HE HAD NO RESPONSE FROM ACR X. AT THIS TIME I ASKED TWR IF I SHOULD SWITCH TO DEP; AND HE RESPONDED 'WHERE ARE YOU?' I TOLD HIM WE WERE AT 3000 FT ON RWY HDG. HE SAID HE DID NOT HAVE ME ON RADAR; SO I CYCLED THE XPONDER. HE THEN SAW ME AND SWITCHED ME TO DEP. I TRIED CALLING DEP 3 TIMES WITH NO RESPONSE. WHILE ALL THIS WAS GOING ON; WE HAD TO INITIATE; TWICE; A CLB BECAUSE OF RA'S ON THE TCASII. WE CLBED A TOTAL OF 1000 FT. THE CAPT THEN CALLED DEP AND GOT A RESPONSE. DEP CLRED US TO TURN TO A HDG OF 350 DEGS TO INTERCEPT THE THUNDER TRANSITION FLL 9 DEP. WE INFORMED DEP WE; TWICE; HAD TO CLB TO MISS TFC. HE SAID HE COULD NOT SEE US WITH ALL THE OTHER CLUTTER AND LATER APOLOGIZED. IT APPEARS FLL TWR 'LOST US.' THEY WERE TRYING TO CLR US FOR TKOF WHEN WE HAD ALREADY TAKEN OFF PER THEIR PREVIOUS CLRNC. I DON'T THINK THEY COORDINATED A HDOF TO DEP EITHER. TCASII TWICE PREVENTED A POTENTIAL DISASTER. I'M NOT SURE WHY TWR WAS CONFUSED.