Air Traffic Controller reported working an aircraft exiting an instrument route that requested IFR handling to their destination but the aircraft was below the Minimum IFR Altitude. The controller instructed the aircraft to maintain terrain obstruction avoidance until reaching the Minimum IFR Altitude for the area.

Narrative

Aircraft X was cleared by ZAB to enter IR137 and exit at point G. Point G is underneath a 160 MIA (Minimum IFR Altitude) and is an alternate exit on IR137. The aircrafts flight plan had point G then direct ZZZ. They exited point G direct ZZZ at about 095; 6500 feet below the MIA. IR137 gives no specification on what altitude to exit at. The leg right before point G is SFC-160. I instructed them to maintain terrain and obstruction avoidance through 160 and climb and maintain FL250; their requested final. Essentially an aircraft on an approved IFR plan left the IR (Instrument Route) route direct destination below the MIA on an ATC clearance. No coordination was done on the aircraft entering the IR route; which is standard procedure from ZAB; so I could do nothing to prevent this situation from occurring.I have no idea who is at fault if anyone in this case. Clarification is requested.- Was ZAB expected to give Aircraft X a cross point G at 160? Despite not knowing the MIA there.- Was Aircraft X expected to know the MIA and exit at 160?- Should an altitude exit be listed on the IR route?- On the IR137 AP/1B page; it states the aircraft is authorized to fly within mission command guidance. Does this mean it is ok to exit below the MIA?- Should the aircraft remain within the confines of IR137 somehow until given a clearance out of it; and is this in conflict with their initial IFR flight plan clearance?- Or is there something else?