A TWIN CESSNA ATX HAS AN NMAC WITH A KING AIR 18 MI NW OF FREEPORT; MYGW; FO

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I WAS ENRTE FROM TREASURE CAY; BAHAMAS (MYAT); TO WEST PALM BEACH; FL (PBI). I CONTACTED MYGF APCH CTL 16 NM E AND TOLD THEM THAT I WANTED TO PASS N OF THE FIELD AT 4500 FT WBOUND. THEY TOLD ME TO RPT ABEAM TO THE N; WHICH I DID. I HEARD THEM CLR A KING AIR IN FROM THE NW (I THINK THAT IT WAS JAKEL INTXN). THEY DID NOT MENTION TFC TO EITHER OF US; SO I ASKED FOR THE KING AIR'S POS. THE CTLR CALLED THE KING AIR AND HE RPTED 18 MI NW. SHE (THE CTLR) SAID THAT HE WAS DSNDING FROM 5500 FT TO 1500 FT. I GAVE MY POS 3 MORE TIMES; AT FREQUENT INTERVALS (I WAS 10 MI NW; WBOUND). I WAS UNCOMFORTABLE WITH THE SIT. THE CTLR SEEMED NOT TO CARE AT ALL (AS USUAL AT MYGF). STILL NO VISUAL CONTACT. I ASKED FOR THE OTHER ACFT'S POS. SHE HANDED HIM OFF TO THE TWR (STILL ABOVE MY ALT). I THEN SAW A DOT TO THE N; ABOVE. THERE WAS NO APPARENT CHANGE IN RELATIVE BEARING. I BANKED HARD L; TO LOAD THE WINGS AND PUSHED THE NOSE OVER. THE DOT CHANGED; IN JUST A MOMENT; TO A WINDOW FULL OF KING AIR. WHEN HE WAS TURNED OVER TO THE TWR (JUST BEFORE I SAW HIM); I SWITCHED #2 COM TO THE TWR FREQ 118.5. I ASKED HIS POS; BUT MY XMITTER WAS STILL ON COM #1. AFTER THE NEAR MISS; THE TWR CTLR NOTIFIED HIM THAT THERE WAS A TWIN CESSNA. THE PLT REPLIED 'YEAH; HE WAS RIGHT IN FRONT OF ME.