I wonder if the Arduino background program is causing the problem? It
does something to keep ocntrol of hte machine. Does anyone know what
that is? I don't.
The reason I ask is that you have no method of turning off the display
entirely. Your program changes one bit of the anode select at a time,
and one bit of the cathode select at a time. It is posible that the
clock spends some time displaying the wrong cathode or the wrong anode,
and this causes the ghost.
You could try to remove U1, the CD4028 anode decoder, and drive all six
anodes directly from the computer. That way, the code only changes one
bit to enable that digit's display.
Alternatively,you just have too much wire between the circuits and hte
tubes.
Please send a photo of the clock.