First thought was fly the aircraft and execute the airspeed unreliable checklist.
Second was at first a guess of a probe problem, as several 737 NG's suffered from unreliable airspeed indications due that in recent events, for iced up probes, but having landed in Mediterranean warm weather and been on ground for ~30 minutes very unlikely. Recently there was even a probe wiring found to be a problem on 737 NG until reversing the wiring sorted the problem. We had been provided background information on these incidents of RYR and NAS both suffering similar issues.
But as AOA was not probe related, we dismissed the probe theory. Altimeter correctness was established using Radio altimeter indicating correctly on operative side and standby instruments.
No checklist for A/T Lim nor when it changed to - - - so maintaining level flight by now using autopilot on functioning side but no auto throttle, a quick reference to FCOM 2 revealed reason and we then concluded from that the possibility of partial ADIRU failure initially as it was the only common source. Later on it failed and recovered afterwards again.
Option to return to field were assessed but as well over maximum landing weight flight continued towards a reduced cruise level and thus destination buying us time and reducing weight en-route. Plenty of destinations en-route to (should need require) divert into.
An airspeed unreliable event does not require a return to nearest alternate, it can be dealt with using pitch and power settings from FCOM-PI-section.
Just remember to declare negative RVSM and set transponder ATC- and ALT-switches to operative side for TCAS.