You know YC's question "How have you ever hacked a non-technical system?" This is my answer to it. You can start a side business on a regular work visa if the character of your business matches your status of residence and you don't intensify the business to the point where it is undeniable that it is a business as opposed to employment, e.g. by hiring full-time employees.
So, for example, if you were an engineer, and you happened to run a software company on the side, you could convince the Ministry of Justice to accept the argument that you were employed as an engineer by the 個人事業 (sole proprietorship) of you, which (since you have an engineering status of residence) doesn't require their pre-approval, and then (while establishing a record of e.g. tax paying and similar) bootstrap that into a renewal of your status of residence with yourself as the sponsor, and from there roll on to either a management/investor status of residence or one of the other more stable options.
For totally-not-a-lawyer gloss on this I'm happy to chat with anyone; you can make reasonable assumptions as to why I am fairly confident that the MoJ can be convinced to allow this.