Technically not a bug. Just an incompatible system.
Find and open Skeleton.shader in your project. It should be under spine-unity/Shaders
You probably want to backup, or only do the edit on a duplicate shader that's named differently. So copy it and save it as a different file.
Open the shader code, then look for all instances of ZWrite and make sure it says ZWrite On
Remember to rename the shader's path on top of the shader code from Spine/Skeleton to something else. Maybe Spine/Skeleton ZWrite
The shader is already under the transparent queue so it should be 3000 by default.
Note that ZWrite means you may have to adjust the Z Spacing slider on your SkeletonAnimation/SkeletonRenderer. It's under the Advanced foldout box in its inspector. Otherwise, you will get Z-fighting.
Also note that using depth also means you'll have hard edges like normal 3D models, unless you have a screen space antialiasing.
You're not actually required to use Spine's shader. It doesn't do anything special. It's just there so users can use something by default.
You could use the Unlit/Transparent Cutout shader if you exported your textures from Spine with Premultiply Alpha unchecked.
For more on that, read this page: Premultiply Alpha
The depth writing is also a standard part of ToddRivers' shader: Sprite Shaders for Unity
You only need to check the checkbox. Don't be intimidated by all the extra features. It acts like the Unity Standard Shader in that if you don't use the feature, it won't be part of the actual compiled shader at runtime so it will run without the extra cost.
