Do I need a TEFL qualification to work in a school?

Hello there. I've done a lot of research but can't find any relevant information specific to what I want to do.

I am a native speaker from the UK with a firm grasp of English, I have a bachelor's degree and can read/write and speak  N3-N2 level Japanese but I have no kind of teaching qualification. I can't find many jobs postings for schools so I am not sure what people want, some jobs for international schools ask for teaching experience specifically, some ask for TEFL/CELTA qualifications, some ask for both or neither, but none specify which TEFL course they want. As I understand it there are varying courses you can do at different hourly amounts, and some of them have opportunity to practice teaching while others don't.

I am not really interested so much in teaching adults at a private eikaiwa-type institute, I would really love to work in a public school. Working as an ALT alongside a teacher is my aim.

Thanks so much for reading, any advice is appreciated.