It looks like the interface is called XAPS and someone was able to build a working implementation last year: http://www.dovecot.org/list/dovecot/2014-September/097721.ht...

From skimming the notes it looks like it's essentially a way to send APNS notifications to Mail.app and requires a cert from Apple. You could run it from your VPS once you get the cert.

NuevaSync has supported this for years, via ActiveSync. https://www.nuevasync.com/nuevasync-pemail.html

Disclosure: have family connection to owners.

Well, wireshark + decompilation of existing software.

What I imagine, though, is a system similar to GCM, which would be not very helpful.

I guess that this is achieved using a server-side communication: the Fastmail server send the notification to the Apple servers and the Apple server sends the notification to the user's device. So wireshark probably won't help.

yeah, as I speculated, similar to GCM. Sadly, there isn’t an open standard for this.