Trying to re-sync Messages between iPhone and Mac

This morning my phone asked me, unbidden to log into my Apple ID (or iCloud, I can't keep those straight). My login did not work (I used the correct password), but I was able to login after resetting the password.

Now Messages.app conversations that I reply to on my Mac do not sync to my phone. My friends got text that said "[my email address] added [my email address to the conversation." and then "[my email address] removed Dwight from the conversation"

How can I get my phone and computer back in sync?

Solution 1:

Well, after about an hour, I was asked (on my mac) if I wanted to add my cell number to Messages. Which I did.