The line runs up through the floor of the house, into the wall, out through the outside of the house wall, which is inside the mudroom, which is attached to the house.
It then runs up the inside of the mudroom roof and into the barn, which is attached to the mudroom, through a hole.
Everything's connected and entirely inside.
Definitely the cleanest thing to do is to remove the old wire and panel from the barn, replacing it with a single, larger gauge wire and new panel.
I'd need to find out if conduit is required or if I can just stape it up to the inside of the structure, as it is currently. That's how all the house wiring is done, but I don't know if it's allowed in an "occupied" space, such as the mudroom, if that's what it's considered.