A B737 crew reported a near miss with an unidentified aircraft as they approached MTPP in a night time non radar environment. The fuel critical flight diverted to a domestic airport.

Narrative

Miami Center handed us off at JOSES intersection and we were told to contact Port Au Prince Center on 124.5. After several attempts we were told to descend from FL330 to FL290 and to fly A315 to GANIV intersection where we would expect to hold. After passing JOSES; we turned on all our exterior lights. Prior to entering the hold we made several more attempts to contact Center to let them know we were fuel critical and would not be able to hold for more than one turn. Communication with Center was very sporadic. At times we could hear talking to other aircraft with no response back and other aircraft trying to contact center with no response back which made it very difficult to keep track of where other aircraft were. We were able to talk to Tower on 118.3 while in the hold to let them know we were fuel critical and requested a landing sequence. They told us there was an airplane on a six mile final and we could be next but they could not clear us to descend. After several more attempts with Center explaining our landing sequence and fuel situation they cleared us to FL240 and then told us to fly the PAP VOR. After prompting them for a clearance after we reached the VOR (which was only six miles away) we were told to fly the 108 degree radial and they would take us outbound for a descent. After the VOR we were cleared to FL170 and soon after to FL150. At the time the flaps were up and the airspeed was 220 KTS. At the start of the descent we received an RA with an indication to climb at a minimum rate of about 1900 FT per minute. Prior to the RA there was never an indication of an aircraft in our location on the TCAS and we never received a TA indication. We never had a visual on the aircraft. We initiated an immediate climb with maximum thrust and the airplane started to buffet. As the back pressure on the yoke was released the buffeting stopped and again we continued to increase our pitch attitude to comply with the RA. The RA ceased after passing FL170 and we leveled off at FL180. We informed ATC of what happened; what our present altitude was and requested an immediate turn and clearance back to ZMA airspace. They cleared us to the VOR and to fly the 340 degree radial outbound. Approaching the VOR we received a TA alert indicating an aircraft 300 FT below us approximately on the 340 radial. We did not turn and the aircraft passed off or right side; we never saw the actual aircraft. We informed Center of the TA and were told now to fly the 012 radial without further instructions. After a few miles we requested a climb and a turn toward JOSES intersection. They told us later after the 35 DME fix we were cleared to JOSES and to climb to FL280. We contacted MIA Center prior to JOSES and got our clearance to our diversion airport and a climb to FL340 after JOSES.