A B737 FLT CREW HAS IMPROPER TFC SEPARATION AFTER RECEIVING A TCASII TA WHILE DEPARTING STI.

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TAKING OFF ON RWY 11 AT STI; AT AROUND 1200 FT MSL; WHILE ON AN IFR FLT PLAN; WITH A LCL ALTIMETER SETTING OF 29.94; AND A HEADING OF 113 DEGS MAGNETIC FOR OUR ACFT AND APPROX A 260 DEG HDG FOR THE OTHER ACFT. WE WERE CLBING OUT AND THE OTHER ACFT WAS DSNDING FOR LNDG; WHEN WE GOT AN AURAL AND VISUAL 'TFC; TFC' ALERT FROM TCASII. WE WERE IN BROKEN CLOUDS; AND DID NOT SEE THE OTHER ACFT IMMEDIATELY; BUT I CHANGED THE SCALE ON MY DISPLAY; AND SAW THE TFC DISPLAYED AT AROUND OUR 12 O'CLOCK POS; AND AROUND 2 NM AHEAD OF US; WITH RELATIVE MOTION OF THE TFC COMING OUR WAY. THE CLOSEST POINT OF APCH; AS BEST I COULD DETERMINE FROM THE TCASII DISPLAY; WAS ABOUT 0.5 NM LATERALLY AND ABOUT 200 FT BELOW US. THE INITIAL TCASII DISPLAY SHOWED THE OTHER ACFT ABOUT 100 FT ABOVE US; AND IT FINISHED ABOUT 200 FT BELOW US. WE NEVER RECEIVED AN RA; I SUPPOSE EITHER BECAUSE WE WERE BELOW 1000 FT AGL; OR BECAUSE OUR PLANES WERE CLBING/DSNDING. NO EVASIVE ACTION WAS TAKEN BY US; AND NO INJURIES WERE SUSTAINED. I DON'T BELIEVE THE OTHER ACFT EVEN SAW US. WE OBTAINED VISUAL CONTACT WITH THE OTHER ACFT (AFTER THE EVENT) AT ABOUT 3:30-4 O'CLOCK LOW POS; ABOUT 1.5 NM AWAY. CALL SIGN AND TYPE OF THE OTHER ACFT (AS RPTED BY SANTO DOMINGO APCH CTL AND SANTIAGO TWR) AND TYPE WAS AN L410 (I DON'T KNOW IF THAT IS CORRECT; IT LOOKED A LOT LIKE AN MU2). ATC SAID TO US THAT 'THE OTHER PLT MADE A MISTAKE -- HE DIDN'T RPT HIS POS PROPERLY TO US.' APPARENTLY NOT! ATC DID NOT SAY IF THEY WERE TAKING ANY ACTION; SO I AM FILING THIS RPT IN THE INTERESTS OF FURTHERING SAFETY.