A family member built my tack box as a Christmas present.... So it's basically custom. But, it looks like a regular square tack box on the bottom, and on the side, it goes up into a "locker" type thing, where I store my bridles. It's quite handy.
Otherwise, you could always just carefully lay it across the top of everything, and then every time it's open, hang it up somewhere. Might be more difficult....
But bridles really do fold up pretty small if need be. Can be bad to the leather though if it's not well conditioned, soft and flexible. A hard leather, might not take to being stuffed in a small place to well.
A large locker might be a little more effective. You can always hang hooks or something in it for your bridles.
I'm trying to find a photo of mine to show.
"You're just as sane as I am."~Luna Lovegood.