IEC 60364-4-41-2005

411.5 TT system
411.5.1 All exposed-conductive-parts collectively protected by the same protective device
shall be connected by the protective conductors to an earth electrode common to all those
parts. Where several protective devices are utilized in series, this requirement applies
separately to all the exposed-conductive-parts protected by each device.

What would happen if exposed-conductive-parts collectively protected by the same protective device be connected to different earth electrode?

You will see "ground shift" between them. "Earth" is not "zero volts" and you can see differences between ground rods that are surprisingly close together. That is why we always bond all the electrodes in a system.

Greg,
I agree with the ground shift thing.
IMO, you are wasting time and materials installing Earth Electrodes any closer than about 12' apart.

One thing I have had good results with in the past, is instead of actually driving a second electrode, is clamping some stripped heavy copper wire to the electrode and burying that copper in a long thin trench at about 2' deep, this has bought down the actual earth electrode resistance and hence increased it's current-carrying capacity under fault current conditions.