Android, sideloading applications and keep them up to date via Google Play

As per January 2015, this appears to be impossible. Here is the response I got from Google Play Developer support:

Side-loaded apps do not update via the Play Store. If you would like the app to update, you will need to uninstall the app on your phone and reinstall the app via the Play Store.

I wonder if anyone found a way around this?

You may install Titanium Backup if your phone is rooted, then long press on your side loaded app package and select "Attach to market" option. Now you may receive updates via Google Play.
I don't recommend this, I mean never install cracked or patched apps because of copyright or security risks, but installing a Modded version of Google Play can make patched side loaded apps to be upgraded after attaching them to market via Titanium Backup. This is not needed for purchased apps because they are registered in your Google account, so attaching to market will be enough.

To summarize the answer from @Android-Developer and OP's comment discussion:

Google Play recognizes the app signature of APKs regardless of the installation source. As long as you use the same package name and keystore when generating your APK, Google Play will be able to detect newer versions and trigger an update.

To install the app without logging in to a Google account, enable the "Install from unknown sources" option in Settings. You can disable it once the installation is complete.