Glitch in Shader with World-Relative Alpha

I attempted to make a shader whose transparency was based on world position. However, whenever alpha:blend or alpha:fade are enabled, I get weird glitching on the camera. Also, even when a pixel should be completely visible, some faces are partially see-through. I'm using a cube with the standard checker pattern, so that shouldn't be an issue

I show the effect of the shader and the glitches in this video:

https://www.youtube.com/watch?v=1lG0QYbHBMg

In addition, at some viewing angles in the viewport the cube is eclipsed by the plane behind it:

My shader code is:

 Shader "Example/Something" {
     Properties{
         _MainTex("Texture", 2D) = "white" {}
     }
     SubShader{
         Tags{ "RenderType" = "Opaque" }
         Cull Off
         CGPROGRAM
         #pragma surface surf Lambert keepalpha alpha:fade
         struct Input {
             float2 uv_MainTex;
             float3 worldPos;
         };
         sampler2D _MainTex;
         void surf(Input IN, inout SurfaceOutput o) {
             float4 c = tex2D(_MainTex, IN.uv_MainTex);
             o.Albedo = c.rbg;
             float a = IN.worldPos.x;
             o.Alpha = saturate(a);
         }
         ENDCG
     }
 }