I don't see a CSS class that is referenced by the guides for things you already call relating to the styling of the popup boxes when hovering over items like recruiting links.

I would like to be able to set the background color, text color, and add padding to this.

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Six Demon Bag
Jack Burton: Hey, what more can a guy ask for?
Egg Shen: Oh, a six-demon bag!
Jack Burton: Terrific, a six-demon bag. Sensational. What's in it, Egg?
Egg Shen: Wind, fire, all that kind of thing!

The recruiting menu currently uses some non-classed popup. I'll have to update it to use the more usefil transtooltip() call I've been using.

For those type of popups, you can override the default with a .transtooltip {} css rule, though you'll have to use a few !importants to make it work.

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It's all in the reflexes.