Why doesn't the Apple TV show up in my iOS AirPlay options after waking up from sleep?

I'm seeing the following after my 3rd generation Apple TV wakes up from sleep (pressing any button on the remote): When I try to connect from an iOS device (iPad or iPhone) using AirPlay, I can't see the Apple TV in the list of available devices.

The options I can see are

• iPad (the device itself)


• My Airport Express

Once I reboot the Apple TV (turning on the TV, going to Settings > General > Reboot) and waiting for the device to come up again, I can see it in the AirPlay options.

Turning off/on WiFi on the iOS device does not help either, it's as if the Apple TV doesn't announce itself properly after waking up from sleep.

The Apple TV is connected to the Airport Express using an Ethernet cable, and it's updated to the latest firmware.

Are other people seeing the same issue?