In electronics, it is customary to define the potential of ground (thinking the Earth as a large conductor) as zero. Is this consistent with the fact that the Earth has a net electric charge that is not zero?
Yes it is, for the following reasons.
In any circuit, you can add an arbitrary voltage offset to the ground return ("lifting" the return above ground voltage) and upon doing the math you will find that the offset voltage drops out everywhere in the equations and leaves you with the same result as the zero offset case. This means that even though the surface of the earth may possess some net voltage offset because of a charging process that is driven by the sun, we can still designate the surface of the earth to be at "ground potential" and all our circuits will still function properly.
By the way, the fact that adding a voltage offset like this is found to have no effect on an electrical system is the result of a mathematical property called electromagnetic gauge invariance.