Same problem, wiped and restored phone and it fixed it temporarily (just like restarting or toggling airplane mode would). Only seems to affect my corporate Exchange account, as I have my Gmail account added as an Exchange account to take advantage of Push and Contact syncing and it continues to work fine. I tested with another iPhone 5 in our organization and it has been working fine since the user got it.
However, I don't see APSD in Diagnostic & Usage Data, I'm not sure how long that log is kept but it has stopped working twice since I performed a restore on the phone, last night and again this morning (only a few minutes after I restarted).
Version 6.0.1 (10A525)
Modem Firmware: 1.01.00
Am going to try wiping and setting it up as a new phone.
Unfortunately, setting it up as a new phone does not seem to have worked for me. Gmail push is still working, corporate Exchange is not. It's been less than 3 hours since I re-set it up as a new phone.
We're using Exchange 2007 in a 2k3 domain with a separate CAS. Doesn't seem to make a difference if I'm on Verizon LTE, corporate wireless (HP and Aruba WAPs), home wireless or VPN'd in.
(Edit: There's a 41 page thread about this issue dating back to September and there doesn't appear to be a permanent fix yet: https://discussions.apple.com/thread/4345374?start=600&tstart=0)