GA pilot reported receiving a request to call ATC after the flight. The call revealed ATC had receive a low altitude alert during their departure procedure. The pilots believed they had performed the departure procedure correctly and safely. In the final discussion; the Controller and pilots agreed the procedure may need to be rewritten to make it clear and easier to understand.

Narrative

Out of RNM; following RNM obstacle DEP PROC Runway 09. Tower requested switch to TRACON. Upon switching; sounded like female voice asked for ident. immediate ident broadcast; and no other transmission rec'd for Aircraft X. Reached 4k and started turn to hdg 330 when TRACON male voice indicated he rec'd a low altitude alert. Advised we were just beginning turn; and was given new hdg to 270. moments later; given new altitude of 5k; and ask to call TRACON as soon as practical. Remainder of flight unremarkable. Mild to moderate turbulence from takeoff to FAF at ZZZ.Phone call to TRACON; spoke to (male) Controller. Indicated the RNM obstacle DEP Runway 09 is 'watched' by TRACON very closely and it appeared we were not turning; requiring Controller to provide vectored deviations. Controller stated they have had many problems with this particular obstacle departure. Stated call was a learning tool for pilots; and he would file a report.Reviewed the obstacle departure procedure on the ground (again) and ADS-B data. We maintained a greater than 500 fpm climb; on the designated course (088) in medium turbulence; and began turn the moment we reached 4000 feet MSL. We believe the Controller may have changed positions and found us at that juncture; and was alerted and changed our course out of caution for our safety; based on previous issues (as the Controller had stated). We appreciate the concern and assurance of our safety by this Controller. We and the Controller agreed that this obstacle procedure requires updating to a safe and easy to understand format and route.This was both Pilots first experience using this RNM Runway 09 obstacle departure procedure and although felt we were inside the confines of the procedure; have now a heightened understanding of ATC focus regarding this departure; and most probably if flown again; request voca (over RNM); cross RNM at or above 5000 before proceeding on course; which would completely avoid the lower level terrain to the east.