nexus - 2 step verification - looks fine, but getting "couldn't sign in" error

A

AC Question

I've tried to enable 2 step verification on my nexus phone - everything looks fine under "accounts->google" I can see that everything is sync'ing ok. The problem is that every 5 minutes when it syncs the mail, it pops up an error "couldn't sign in" (to Gmail) in the notifications. This is strange because gmail is sending/receiving ok.

I've narrowed this down to the fact that my Nexus seems to handle email a bit funny. I have a gmail account AND a corporate account, which means I have a gmail app and a "mail" app that seems to have both the corporate email and the gmail. This is the sucker that seems to be throwing the error. When I click on the "couldn't sign in" it pumps me to the "account settings" page where you can select incoming server setting etc., but never prompts for the the two setup authentication (which I previously, and successfully setup on my gmail).

Any hints on how to turn this off, or how to get the corporate email only on the "email app" and the gmail only on the "Gmail app"?

Thanks!