I already know how to do this theoretically. I have to draw the object twice:
Once with normal settings and
once with a different depth sorting mode where only pixels that are behind rendered geometry are shown. Also, to prevent weird effects later on, these pixels can't be rendered into the depth buffer.
I can set the correct depth sorting mode in Stage3D with Context3DCompareMode.GREATER. Is it also possible to have Stage3D render pixels into the back buffer, but not the z buffer? If I can't keep Stage3D from rendering to the depth buffer, the effect will look like this: