Now I don’t speak any Japanese but this Japan site with tips on Onsen bathing (in German) includes a link to a site with a directory of tattoo-friendly spas/baths/onsen. Since it is in Japanese only and I know next to nothing about that I can’t judge for its content, but it does seem very legit.
There are a variety of grades of public bath in Japan, reflected in the facilities and the pricing.
Super sentos and up normally prohibit tattoos. Every sento I have been in has had at least one person with tattoos, and sometimes the saunas are obviously being used for business meetings by people with punch perms who describe their jobs as “a little bit of this and that.” Those places are more interesting anyway.
There are lots of public baths in Japan that are free, unattended and thus open to everybody. Here are a few in Yagen Valley, Aomori, built right next to a river:
And I shared the first of those with some gentlemen with full-back tattoos, done in a traditional style instantly recognizable to any Japanese.
Now Yagen’s about as far from it all as it gets in Japan, but there are lots of these in rural areas if you look, some of them quite famous (Jigokudani of monkey fame, Kawayu Onsen, etc).
And not all professionally run onsen are anti-tattoo, here’s a good (Japanese-only) listing of tattoo-friendly onsen in Hokkaido. Interestingly, quite a few say things like “OK as long as it’s not full-back” or even “Fashion tattoos are OK, but irezumi [Japanese traditional tattoos] are not”!
Many hotels and ryokan have a family bath, that is, a bathing facility that can be reserved for private use by families or other groups. This can range from a simple bathroom with a large tub available for free to guests of the hotel up to an luxurious outdoor onsen at $50 per hour.
Of course, you would be alone there unless you bring along friends. I can’t tell from your question, but if by “experience of a public bath in Japan” you mean that you want to mingle with regular Japanese guests, you’re probably out of luck; the ban on tattoos is not specific to publically owned facilities; private operators are just as (if not even more) likely to have it.
There are baths that are run by or are friendly to the yakuza and also open to the public; those will allow you to enter with tattoos.
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