gravity - Why can't General Relativity be written in terms of physical variables?

I am aware that the field in General Relativity (the metric, $g_{\mu\nu}$) is not completely physical, as two metrics which are related by a diffeomorphism (~ a change in coordinates) are physically equivalent. This is similar to the fact that the vector potential in electromagnetism ($A_\mu$) is not physical. In electromagnetism, the equations can be written in terms of physical (i.e. gauge invariant) quantities -- the electric and magnetic fields. Why can't Einstein's equations similarly be written in terms of physical variables? Is it just that nobody has been able to do so, or is there some theorem/argument saying that it can't be done?

EDIT: Let me rephrase: Prove/argue that there is no explicit prescription that can be given which would uniquely fix coordinates for arbitrary physical spacetimes. I.e., show that there is no way to fix gauge in the full theory of general relativity (unlike in E&M or linearized GR where gauge can be fixed).