Yes, I do. It is entirely possible for a foreigner to become a full-time teacher in a Japanese school. The catch is that a foreign teacher must pass the same testing and licensing examinations as a Japanese teacher.

A few years ago a couple prefectural boards of education became tired of dispatch company ALT services and began a program to hire foreign teachers directly. These boards of education also provided training which would allow these foreign teachers to obtain formal Japanese teaching certifications.

The main reason behind the part-time requirement for ALTs and other teachers has less to do with the teacher's nationality or citizenship than it does with schools and dispatch companies not wanting to pay for the more comprehensive benefits which come with full-time work.

So am I mistaken in thinking that public school teachers are koumuin, and to be a koumuin you must be a Japanese citizen?

Yes. When I was in JET my boss was the Hyogo Prefecture Board of Education (not the school) so I was a public employee and, obviously, a foreigner.

I'm obviously not talking about JETs and other ALTs, who are usually hired for a limited time period. There seems to be some sort of exception made for JETs and direct BOE hires- I am talking about a permanent employee, as in a foreigner becoming a geography or maths teacher in a public school (let's assume their Japanese is fluent).

Have you ever heard of anyone who's done it without becoming a Japanese citizen?

Once again, yes I have.