EEVEE implements SSAO or generic AO?
Anyway, HDRI casting shadows was proposed somewhere else. Something like that was possible in the past via scripts that placed some lights (similar to non traytraced AO effect) for strongest parts of image, but it could be automatic now; it will be harder with EEVEE due to low limits. Those will need to be raised, or a completly different way found (lighting pre-pass(es)?).