Why does my opaque texture seem to be transparent?

I'm having a strange issue where I have set all objects to use a standard opaque material (with their own textures) but when two objects are overlapping, they appear to be transparent. That is, the two textures are simultaneously visible and seem to be combined.

This is the material in question (the texture is applied in code):

Below are 2 overlapping objects with the same texture. The darker diamond shape is where there is NO overlap. The 'brighter' areas are where the two objects overlap. I don't know if it matters, but the objects are custom meshes (rather than sprites).

My question is, how can I make the textures fully opaque (such that you would not see the diamond)? I tried changing the shader to various other values, but doing so caused the shadows cast and received by the objects to disappear.

UPDATE 1

Here is the function which constructs the object mesh (note that texture and material are set prior to this call. texture is of type Texture2D ):

     override public void Construct() {
         mesh.Clear();
         float halfHeight = Mathf.Sqrt (0.75f);
         Vector3[] Vertices = {
             new Vector3( 0f,    0f,         0f),//Z //FRONT
             new Vector3( 1f,    0f,         0f),//A
             new Vector3( 0.5f, -halfHeight, 0f),//B
             new Vector3(-0.5f, -halfHeight, 0f),//C
             new Vector3(-1f,    0f,         0f),//D
             new Vector3(-0.5f,  halfHeight, 0f),//E
             new Vector3( 0.5f,  halfHeight, 0f),//F
             new Vector3( 0f,    0f,         0f),//Z //BACK
             new Vector3( 1f,    0f,         0f),//A
             new Vector3( 0.5f, -halfHeight, 0f),//B
             new Vector3(-0.5f, -halfHeight, 0f),//C
             new Vector3(-1f,    0f,         0f),//D
             new Vector3(-0.5f,  halfHeight, 0f),//E
             new Vector3( 0.5f,  halfHeight, 0f) //F
         };
 
 
         Vector2 scale = new Vector2 (0.5f, 0.5f);
         
         float unitWidth = scale.x;
         float unitHeight = scale.y;// * Mathf.Sqrt (0.75f);
         float unitHalfWidth = scale.x * 0.5f;
         Vector2 FZ = new Vector2 (offset.x, offset.y);
         Vector2 FA = new Vector2 (offset.x + unitWidth, offset.y);
         Vector2 FB = new Vector2 (offset.x + unitHalfWidth, offset.y - unitHeight);
         Vector2 FC = new Vector2 (offset.x - unitHalfWidth, offset.y - unitHeight);
         Vector2 FD = new Vector2 (offset.x - unitWidth, offset.y);
         Vector2 FE = new Vector2 (offset.x - unitHalfWidth, offset.y + unitHeight);
         Vector2 FF = new Vector2 (offset.x + unitHalfWidth, offset.y + unitHeight);
 
         Vector2[] UV = {
             FZ, FA, FB, FC, FD, FE, FF,
             FZ, FA, FB, FC, FD, FE, FF
         };
 
 
         int[] Triangles = {
             0,1,2,0,2,3,0,3,4,0,4,5,0,5,6,0,6,1,
             7,9,8,7,10,9,7,11,10,7,12,11,7,13,12,7,8,13
         };
 
         Vector3[] Normals = new Vector3[14];
         for (int i=0; i<=14; i+=3) {
             Vector3 left = Vertices[Triangles[i+1]] - Vertices[Triangles[i]];
             Vector3 right = Vertices[Triangles[i+2]] - Vertices[Triangles[i]];
             Normals[i] = Vector3.Cross(left, right);
         }
 
         GetComponent<MeshFilter>().mesh = mesh;
         mesh.vertices = Vertices;
         mesh.uv = UV;
         mesh.normals = Normals;
         mesh.triangles = Triangles;
 
         MeshFilter filter = GetComponent<MeshFilter>();
         filter.mesh = mesh;
 
         mesh.RecalculateBounds ();
         mesh.RecalculateNormals ();
         mesh.Optimize ();
 
         MeshCollider mc = gameObject.GetComponent<MeshCollider> ();
         mc.sharedMesh = mesh;
         
         GetComponent<Renderer>().material = material;
         GetComponent<Renderer>().material.SetTexture ("_MainTex", texture);
 
     }

UPDATE 2

I checked and the material.color is set to RGBA(1.0,1.0,1.0,1.0) which makes this even more baffling.

Could lighting be causing this issue?