Air carrier Captain reported responding to TCAS RA resulted in a NMAC event while at cruise altitude in Cairo (HECC) airspace.

Narrative

At Cruise FL350; 52 miles N of DATOK; L550 southbound; Cairo airspace RA event with Aircraft Y. At approximately XA:30 Z. We heard increased urgent transmissions on Cairo frequency in a foreign language. Looked outside after seeing opposite direction traffic 200 above and climbing as indicated on the TCAS. We Decreased the scale on Navigation Display to ascertain exact location of traffic. Then received a traffic advisory Traffic Traffic;" one time. This was immediately followed by "Descend Descend." Executed RA maneuver disconnecting the Autopilot and pushing over. Traffic passed high and to the left. We bottomed out at 34;200; and the RA ceased. We informed the controller we had a RA and climbed back to FL 350. Upon request the controller informed us the other aircraft call sign was Aircraft Y. Additionally; the controller informed the other aircraft was confused about their clearance. Aircraft Y is also call signed Aircraft Y. The TCAS worked well however there was little warning of the TA and RA. It happened very quickly. Other aircraft altitude deviation; believe your TCAS."