Repair Station Manager reported receiving and installing an incorrect turbine engine due to paperwork inconsistencies for both the aircraft and engine manufacturers.

Narrative

Aircraft had history for slow starting since it was new; Our organization troubleshot per the aircraft and engine manufacturer's recommendations (Note: aircraft is under warranty) which ultimately led to removing original engine which was sent to the engine Mfg for evaluation and installing a rental engine provided by the engine mfg. Ground testing revealed loaner engine met all parameters for takeoff power and aircraft was released to service. Aircraft was returned to owner by a company professional pilot whom stated the aircraft flew normal and reported no anomalies. A while later the owner contacted us and said the aircraft flies slower than the published cruise numbers and later brought the aircraft back to us for a scheduled annual inspection at which time we performed performance runs and noted no power issues on the ground. We did recalibrate the auto throttle system thinking that may have been a factor but later the customer reported that it still was flying slower than published. On DATE we were notified by Name (A/C Mfg) and Name 2 and Name 3 (Eng MFG) that we had installed the wrong build spec engine on the aircraft even though it was the correct model engine which was the cause for it to fly slower. There is no mention of a certain build spec for installation in either the aircraft Type certificate or engine certificate nor is there any mention of it in any of the maintenance publications. The rental engine was reserved and provided by Name 2 and 3(Eng MFG). Our team received an email on DATE1 that rental engine S/N XXXXA had been reserved. The rental engine arrived to our facility on DATE2. We waited to open shipping crate until we received approval from Name 2 and 3 (Eng Mfg.); which was granted on DATE3. On DATE4 at XA:27CST Name 2 and 3 (Eng Mfg) provided email containing cert package for rental engine S/N XXXXA. On DATE4 at XB:43CST we received an email 'Please disregard my previous cert package. I have attached the correct one...'. The updated cert package was for rental engine S/N XXXXB. The original rental engine S/N XXXXB was the correct build spec for the aircraft. The rental engine S/N XXXXA that Name 2 and 3 (Eng Mfg.) had actually shipped was not the correct build spec; and as expressed earlier there is no guidance on any build spec requirements to install PT6A-66D.