OK Bryan so you know about the registering. From what I've seen on the genuine articles in Japan they usually have the register lip on the base. Here's a picture from a book on Japanese furniture showing construction methods for boxes.
Our method of adding extra strips for the registers doesn't figure! It looks like they route separate lips on the base and the tops as Eric suggests, which I think is harder, given that they use very thin stock. 'Our' glued in strip method whether in the top or bottom of the box I think is easier. So, the bottom right in the above picture. As Eric suggests any routing needs to be very accurate - plus you loose a fair bit from the height of the box.
Did you use solid wood or veneered ply for the main box? I've used paulownia a few times which is what I think the Japanese use. It's soft but exceptionally stable, and requires equally exceptional sharp tools to work it successfully! But its hard to come by down here.
As you see in the picture they tend to just glue their small boxes together - no joints. But from what I've seen on their lacquered boxes, they then cover the wood with a type of thin muslin 'cloth' soaked in lacquer - which you've probably seen. Which then needs lots of coats of filler and lacquer to bring to a flat surface.
The Pinterest link worked for me when I copied and pasted it into my browser, but here's the Pinterest pages as a URL link.
