B737-800 First Officer reported engine failure with FADEC and reverser lockout malfunction resulted in an expeditious return to departure airport to an overweight landing.

Narrative

I was the FO assigned to Aircraft X; ZZZ-ZZZ1. I was flying with Captain. We are both ZZZ1 based pilots on the 737. Aircraft X was the last leg of a two-day trip.The sequence was very routine in nature. The inbound flight was delayed coming into ZZZ which obviously delayed our outbound flight. We were flying Aircraft X; which had MEL 49-XXX; for the APU BLEED AIR VALVE. Once we completed our preflight checks and were boarded; we conducted the supplementary procedure per the AOM for engine start. Other than the supplementary start procedure; it was just a routine push back and taxi. It was the captain's leg to fly; and we took off XXL out of ZZZ. Somewhere; around 9000-9500 feet; the captain noticed something wrong with the aircraft. It was at this point that I noticed the indication showing the #2 ENG REV was illuminated; with a loss of thrust on the #2 ENG; and the #2 throttle did not move via auto throttle. I believe the #2 EEC (Electronic Engine Control) took over the #2 ENG thrust when the Reverser indication illuminated; but I am not sure. The captain continued to fly the aircraft while I [requested priority handling] with ZZZ departure; requesting a level off at 10;000 feet and a heading; with an intent to return to ZZZ airport; and requested them to stand by while we figured out the issue and ran some checklists. Over the next few minutes; the #2 ENG REV indication did go away which then was accompanied by an increase in thrust to what was being commanded by the position of the #2 throttle; I believe. Once the #2 ENG spooled up; the #2 ENG REV indication re-illuminated; and accompanying loss of thrust on the #2 ENG. The cycle repeated itself maybe 2 or 3 times. There was a noticeable yaw in the aircraft but was controlled the captain flying. While this was going on I opened the electronic QRH in Comply; I did make a mistake opened the REVERSER 7.XX checklist; but quickly noticed my mistake as that was for ground operations; and then opened the correct REV 7.XY checklist. The captain continued to fly while I read the checklist. This then led us to the Engine Failure or Shutdown checklist. It was around this time we began a decent to 8000 feet I believe. I then ran the Engine Failure or Shutdown checklist which we completed for the #2 ENG. Once completed; I then ran the One Engine Inoperative Landing checklist; referencing how much landing distance would be required at which point I noted that it would additionally be an overweight landing. Once these were completed prior to the Non-Routine Landing Considerations checklist being completed; the captain and I swapped pilot flying duties vs pilot monitoring duties. The captain then notified dispatch of the issue; the flight attendants and the passengers on the situation and that we would be returning to ZZZ. During this time; I also began set up and reviewing for an approach to ILS XXC back into ZZZ. It was around this time I stated to ATC; Fuel on board; Passengers on board and the nature of the [situation] as well as stating we could begin our arrival into ZZZ. ATC gave us headings to complete a 360 degree turn prior joining XXC LOC. It was around this time to the best of my recollection that the captain requested the aircraft back; and we swapped pilot flying vs pilot monitoring duties. I then ran the Non-Routine Landing Considerations checklist; as well as the Overweight Landing checklist. Once completed; we reviewed the approach plate for ILS XXC into ZZZ; as well as running the Before Landing Checklist. It was around this time we intercepted the XXC course for landing and began our approach. Once closer to the runway; I noticed ARFF (Airport Rescue and Firefighting] and emergency response vehicles parked on the adjacent taxiways to XXC which then followed our aircraft after exiting the runway taxiway 1 off XXC. Once stopped; the captain requested the passengers remain seated while I communicated to ARFF command; requesting an inspection of the #2 ENG for any abnormalities. Once; the inspection was completed and they stated they didn't see any abnormalities; we cancelled out [priority handling] status. I then coordinated with ZZZ Company operations for an arrival gate of XX. The captain then began to taxi the aircraft towards XX with ZZZ ground instructions while I called the flights attendants and confirmed everyone was ok and that we would be going to Gate XX. Once; at the gate we completed the shutdown checklist. The captain made a logbook entry for the incident. He then spoke to the director. At which point we then booked tickets on the next ZZZ-ZZZ1 flight home as we had already gone over our over time for the day. This statement is the best recollection I have of the events as they happened for Aircraft X. I do believe the crew; as a whole; did an amazing job and I was fortunate to have completed my annual recurrent training recently; which made me feel that I had recency in dealing with [a situation] of this type. I do believe my Company has a great training program with knowledgeable individuals coming together to train us for when a situation like this arises. No recommendations. Company has a great training program.