Will be going to Japan soon with my GF. She fancies going to an onsen sometime during the trip but does not like the idea of gender-separated baths and also she is quite shy so we would try to find either a private onsen bath or a not too crowded free "natural" onsen bath (not sure if that is called onsen though). About natural onsen I know I have seen some on Japan-guide which are a little or not prepared for bathing but are meant for use as onsen, if you see what I mean.
We will be staying in Fukuoka, Osaka, Sapporo and Tokyo for the trip so if there are any okay ones around there good for daytrips then please let me know. Would prefer those that are not needed to have an active stay at a hotel or ryokan near the onsen as I have heard some are required to be a guest of those to be visiting the onsen.