You don't. The coat looks that way because of its sheer volume.
I've made five or six of these coats for commission, including both the white Battle City coat and the purple sleeved version. I always start with a basic flared coat pattern (not a trenchcoat, think more like the Matrix coat pattern) and then add in triangular gores to the side seams. Look up a "gored skirt" to see what that means. It's basically adding triangles of fabric into the seams to add in volume.
If you take the series as a whole, you'll notice that Kaiba's coat does move naturally, in the wind or when he walks fast. It flaps in the wind. So trying to use a wire or boning or anything else to stiffen the side or bottom hem will just make the coat sit unnaturally around you. If you've seen a hoop skirt, and a person walking around in a ballgown, you'll know what I mean. They just float around like a living wedding cake. That isn't natural. If you want Kaiba's coat to look natural and yet still amazing and accurate, it just needs to have huge volume in the bottom. My roommate's Kaiba coat, for example, has a bottom hem that's 17 feet long. I think my Amelda coat is about the same length around the bottom.
Founder and President, Madison Area Costuming Society, a chapter of the International Costumer's Guild
Strange Land Costuming - www.strangelandcostumes.com