Model with transparency

Model with transparency

I have a model with transparent quads for a beard. I can not tell what
triangles belong to the beard because their color comes from the texture
passed to the fragment shader. I have tried to presort the triangles back
to front during export of the model, but this does not help. So I
implemented MSAA and Alpha to Coverage, but this did not help either. My
last attempt was to draw the model, with a depth mask off and skipping any
transparent data, so the color buffer would have non-clear color values to
blend with. Then I would draw the model a second time with depth testing
on and drawing the alpha pieces.
Nothing I have tried so far has worked. What other techniques can I try to
get the beard of the model to properly draw? I am looking for a way to
handle this that doesn't use a bunch of extensions. I'd prefer techniques
that can be handled with plain old OpenGL 4.
Here is an image of what I am dealing with.