PC-12 Captain reported loss of pitot static anti ice systems during cruise and unreliable air data information on both the Captain's and First Officer's air data systems. The flight crew diverted to a nearby airport.

Narrative

During the cruise at flight level 230 in IMC with icing and light chop conditions in PC 12/45; several unusual events occurred as follows. There was no CAWS (Central Advisory & Warning System) message in any of these events. 1. Sudden loss of First Officer's side airspeed indication 2. A few minutes later; suddenly lost the First Officer's side altimeter indication. (Enter rain conditions for about a few minutes.) 3. A few minutes later; after departing in rainy conditions; the Captain's side airspeed indication gradually lost. 4. Upon reaching zero airspeeds on the Captain's side of airspeed indication; experienced control wheel shaking (much lighter shake than stick shaker but no verbal stall warning. I used the pitch/power method to estimate and maintain the speed and altitude.)Contact the Center and request a lower altitude due to the loss of airspeed indicators. The Center provided us with some airports available and we requested a vector to ZZZ as we know there is a [maintenance center] in that field. Radar vector and descent instruction was given and LOST CAPTAIN's side ALTIMETER INDICATION during descending to the assigned altitude. We reported we lost both altimeters as well. We are about to request when to level off based on radar reading from the center; and we got out of the cloud and entered VMC. A few minutes later out of the clouds; the following events occurred at almost the same time. 1. The First Officer's airspeed and altimeter recovered to normal indications. 2. Captain's airspeed and altimeter recovered to normal indications. Reported Center that all indications are normal; but still request to vector to ZZZ to full stop. Enter right downwind; base; and make a full stop landing at Runway XXR.