I’m testing EEVEE and it seems that direct light (e.g. from the sun in an HDRI) will not cast shadows. Maybe there is some trick to get this to work, or is it a limitation of the EEVEE renderer…?
It is a limitation. EEVEE is different from Cycles by the way it handles lights.
A background or an emitting mesh will only have an effect on scene if a light probe (object in charge of caching Indirect Lighting) is used.
But penombra that can be produced using a light probe can not have the quality of a shadow produced by using a Sun lamp.
You will have to use a sun lamp and orient it in the same direction of sun from HDRI.