As you suspect, there is a problem with static electricity.
If I had the globe in my hands, I would be looking at the base and how the mechanics and electricity make it from outside the globe to inside the globe - because that will be your access path.
You objective is two-fold:
1) To create a grounding path between the Styrofoam at the other interior components; and is that is not sufficient
2) To create a grounding path between the interior of the globe and the exterior.
My thoughts are this: A little bit of moisture alone would likely do the trick. So, for a 4-inch globe, perhaps 2cc of water inserted through the base and then mixed (by shaking the globe) with the Styrofoam.
But, if that wasn't good enough, I would want to introduce a dilute solution of
diethylester dimethyl ammonium chloride (aka, DEEDMAC), and I would want that well-mixed with the water I already added. So, I would skip the water-only step and go directly to the DEEDMAC solution.
DEEDMAC is easier to come by than you might expect. It's the main ingredient in fabric softeners. So, you will be looking for a fabric softener with minimal "other stuff" - such as fragrance, stabilizers, coloring, etc. It's not that there's anything wrong with a fragrant snow globe, just that you will be putting a very thin coat of this on the inside of that crystal-clear globe - and you would like to keep it crystal clear.
So, here's what I would do:
1) It shouldn't be necessary to use the DEEDMAC at full concentration. So, I would make up a mixture of about 1 part softener to 3 parts water.
2) I would apply a couple of cc's of this to a clean window pane and let it dry.
3) If the window pane still looked clear enough, your mixture is good. Go to step 5.
4) If not, try a different fabric softener, distilled water, or more dilute concentration and go back to step 1.
5) Measure out about 2cc of this mixture and deposit it into the globe through the base - hopefully through a crack or opening that is already there.
6) Shake it up to spread the DEEDMAC throughout the interior of the globe.
If this fixes it, you are done. If not, up the dose a bit. In the unlikely event that this still doesn't work, you may need to discharge the clear plastic globe itself may be acting like a capacitor - and what you've done so far is only the first step to fix it.