EMB-120 Captain reported an engine fire loop began to indicate erratically on climb. Pneumatic pressure began to fluctuate and cabin altitude could not be controlled. Flight crew performed an air turn back to departure airport.

Narrative

During climb through 12000 ft.; Left Engine 'DET INOP PIPE ZONE' annunciator began flickering. Shortly after defect was noticed; 'CABIN ALT' annunciator and triple chime/Master Warning illuminated/sounded. On cabin pressurization gauge; cabin altitude showed 12000 ft. and differential pressure showed 0.3 psi. I verified that the Controller switch was in AUTO; the pressurization rate knob was at the green band; the pressurization barometer was set correctly; and that the bleeds were on; set to AUTO; and the packs were set to LOW. I notified ATC that we had an indicator malfunction and expected to return to ZZZ1. After notifying Dispatch via radio; I asked for a return to ZZZ1 from ATC and explained that we did not need [special handling]; did not need priority; and were conducting a precautionary return for an indication malfunction that would have grounded the aircraft at the destination. During descent; the DET INOP annunciator continued to illuminate intermittently until descending below 6000 ft. I reset the cabin pressurization to 300 ft. below ZZZ1; and later to 500 ft. below sea level in an attempt to increase demand for differential pressure. During descent; differential never rose above 0.1 psi but ATC kept us descending so there was not enough time to evaluate whether the differential was increasing as commanded. I saw the DET INOP annunciator flickering; and after the CABIN PRESS annunciator illuminated I checked the cabin pressurization gauges. A part of the fire detection system is not working as designed; and the aircraft pressurization system was not providing more than 0.3 psi. I notified ATC and Dispatch; asked for a precautionary return; and attempted to verify that we were operating the pressurization system correctly. Complex systems have many points of possible failure.