Okay, now I am spooked. I was talking to a friend on Google Chat (part of the GMail interface). Through stupid absent-mindedness I typed a question, then closed the GMail tab and went to other things.
A minute later the reply came to my Android phone.
Apparently Google figured out that I was no longer "on" the chat and re-routed the message. Useful, yes. But scary, too! My phone is tied into my Google account, otherwise it'd be pretty useless, but I still try to keep up pretenses by disabling GPS, location services, all that. But there is no escaping anymore.