If you stucco your oven with surface bonding cement you can expect your dome to stay "reasonably" dry. However, that's really not good enough. If your earthen dome should get wet, it really doesn't take much for it to revert back to its natural state, and that would be "mud."
For maximum protection, consider a roof, and it can be as primitive as the roof over the Siberian oven, but by far, the simplest roof I know how to make is to drape corrugated metal roofing over the dome and fasten the four corners with bungee cables tied into the oven base.
For maximum protection, consider a roof, and it can be as primitive as the roof over the Siberian oven, but by far, the simplest roof I know how to make is to drape corrugated metal roofing over the dome and fasten the four corners with bungee cables tied into the oven base.
Oven stuccoed with surface bonding cement
Simple roofing system
Easiest roofing system