It is not possible to apply an affine transformation to the tile and get the result you want, because affine transformations preserve coplanarity, whereas the shape you need is not flat in the 3D world. Even if you got the outline to match (which can be done), the texturing would be all wrong and cause seams at the junctions.

I suggest splitting the tile in two triangles, either way you want, and project/render these triangles separately:

The good thing is that it can be done without any additional calculations. You just need to render twice and put transparency in the unused part of the images. You can then reuse exactly the same transformation as for your light green tiles:

It is not possible to apply an affine transformation to the tile and get the result you want, because affine transformations preserve coplanarity, whereas the shape you need is not flat in the 3D world. Even if you got the outline to match (which can be done), the texturing would be all wrong and cause seams at the junctions.

I suggest splitting the tile in two triangles first. There are two ways to do so, but the first one (vertical split) will probably be easier later on:

The good thing is that it can be done without any additional calculations. You just need to render twice and put transparency in the unused part of the images. You can then reuse exactly the same transformation as for your light green tiles: