GFI trips when turning on a computer with the monitor on but not when the monitor is off

I have an unusual situation here. I have an older Dell computer that is plugged into an outlet. The monitor that I use with this computer has its own power switch and is also plugged into the same outlet.

If I turn on the computer first and then the monitor, all works okay. However, if I turn on the monitor first and then the computer, a GFCI on a different circuit trips. The neutral wire for this particular outlet is not shared anywhere with the wiring of the other affected circuit except, of course, at the box where all the neutrals are shared. There are additional outlets on this circuit where my Dell is plugged into.

A couple of things to note:

1. There are several individual circuits coming from the box and because the box is located in a shed away from the house, every circuit has its own GFCI.
2. Two of the circuits have GFCI breakers while the remaining circuits have GFCI outlets. One of the GFCI breakers is the one that trips.
3. The computer is plugged into a regular outlet downstream from its GFI outlet.
4. The separate wires for each of these circuits do run parallel to each other inside the wall.

Michael