Render texture above something else

In my scene I have a 3D phone with a video on the screen (and overlapping the sides) which has a chroma key shader on it: https://www.assetstore.unity3d.com/en/#!/content/15172

The keying works great but I get some serious Z-battling with the phone underneath, despite the plane being clearly in front of the 3D phone. It gets worse the further you move away from it.

I've tried to change the 'Tags' so that Transparency is +1 (I've tested up to +4) but with no change.

The phone was originally textured with Standard shaders but I've also tested with Mobile/Diffuse, all to no avail.

Can anyone tell me why this is happening and how to fix it?