Even in electromagnetic waves, as they are generated by oscillations of charges, E is not perpedicular to B. What seems to happen here is that the non perpendicular component of these fields to the direction of propagation (Poynting vector) falls more intensely than the perpendicular component. Therefore, far way, we are likely to observe just E and B perpendicular to each other and also to the direction of propagation.
If you set charges in static regime and set currents with other charges you will probably fill the space with E's and B's which are not perpendicular to one another mostly.
best wishes
DaTario