I don't know, this may be one of those things there's no way around. 

I had similar qualms when designing Laughter on The 23rd Floor which requires writing on the wall with a indelible marker AND and punching 3 holes in the wall every night.

Anywho, food coloring may or may not stain less than soda, and tea, without doubt has staining qualities. 

Perhaps you could put a fashionable rug in the target area of darker color quality than the other carpet.  And maybe you could just know that you'll have to be espcially conscientious of that area.  The Mr Clean cleaning bricks work well on wall stains, they remove Sharpie.
:) 

At the end of the day, with tricks as such; sometimes the scenery needs more attention than previously thought.

Just a thought one thing you could try K8TT! Buy a good brand baby?s nappy [dipper], which is super absorbent. Take out the inner polymer liner & put it in the container, then add coke when you need to & it will super absorb the liquid from 150 to 300 times it?s weight instantly!
If this is a hassle making the polymer fibre fit in the cup!
Buy some polymer crystals from a supplier & just put them in the bottom of the cup. What is good about using the crystals is you can dry them out on paper & use them again, also they are cheap!

 Having done a number of slap stick custard pie shows in my time. I sympathise with you K8TT!
Even using removable tension stage cloths, the shaving cream gets every where & can become a very slippery problem. The potato crisp crumbs are worse, we used to use them & corn flakes crunched up on the floor to add sliding for ballroom dancing!
Even if your treat the stage deck to make it anti skid, by adding  high sugar content soda or soft drinks in the mop/wash boiling water. {Diet drinks with chemical sweeteners don?t work as well as the ones with sugar] Food bits act like stepping on lighting gel left laying around on the stage & cause an A over Turkey without thinking!
One thing you could consider is making up a resin box for each side of the stage in the wings. The performers step into them before making an entrance on stage, in the same fashion as used by ballet dancers.
These are just a piece of flat plywood with a small wooden trim around the edge to hold the powdered resin, big enough to place a foot in. If you have a painted black deck or use a dark coloured Tarquete dance mat, you will see where everyone has walked, great for ballet but it might not be your tea bag!
Let us know how it all turns out K8TT!

Would it work for the drink to be almost empty,therefore it'd just be ice water that's thrown?