A B737-800 Flight Crew struggled with recurring erroneous air data information ultimately tracked down to a malfunctioning ADIRU (Air Data Inertial Reference Unit).

Narrative

All was normal during initial climb out out of a tropical area. At approximately FL200; we transitioned from VMC to IMC in a large non-CB cloud that had no significant signature on our RADAR. We experienced light turbulence and did not notice any precipitation. At approximately FL250; 'IAS DISAGREE' was displayed on both PFDs. There were differences (relatively minor) in Captain and First Officer IAS and Mach. We complied with the QRH 'IAS DISAGREE' and 'Flight with unreliable airspeed' checklists. We also consulted the Performance Handbook for Pitch/Power information. We confirmed that the Pitot Probe Heat was ON. By approximately FL280; we transitioned to VMC again with no further IMC in sight ahead of us. The 'IAS DISAGREE' message cleared and all indications became normal again. Our OPT altitude was FL350. Our flight plan altitude was FL330 in this segment. We elected to amend our cleared altitude to FL330 to increase speed margins should the problem re-occur. We contacted Dispatch and Maintenance control for guidance and possible air turn-back. We informed them of what had just happened and that all seemed normal at this time. They suggested we continue towards our intended destination. We concurred. Note that our ETOPS Entry Point was still ahead. The flight proceeded normally with no further anomalies during the cruise phase. As a precaution; we compiled an abbreviated list of pertinent Pitch/Power/VS combinations for our descent should the problem resurface. During our descent; we re-entered IMC passing through FL250. The 'IAS DISAGREE' message was displayed once again on both PFDs and we noted differences between Captain and First Officer IAS and Mach. The differences became significant. We noted initially on PFD's: CAPT IAS=296kt; M 0.74; First Officer IAS=315kt; M=0.77. We noted conflicting information on Nav Display TAS and wind read-outs. We obtained Ground Speed readouts from ATC. We cross-checked other instruments such as the HUD Angle-of-Attack readout. The EECs soon reverted to ALTN. We complied with the QRH and placed the EECs in 'hard ALTN'. The situation led us to DECLARE AN EMERGENCY before reaching 10;000ft. Differences in Captain and First Officer IAS became very significant (Captain IAS 170kt; First Officer 250kt). It was clear that the Captain information was erroneous; therefore controls were transfered to the First Officer. A momentary stick shaker occurred while the First Officer IAS was 240kt with normal Angle-of-Attack indication in the HUD and normal Ground Speed. The approach continued without further incident. We selected a target approach speed of Vref+20kt to account for possible unknowns.At this time; I do not know what caused the Captain ADIRU (Air Data Inertial Reference Unit) to malfunction.