Kyushu Minamioguni, Kumamoto
Ryokan Ichinoi (Kurokawa Onsen)
Name in Japanese. Look for this on the sign, or show it to a taxi driver.
Onsen ryokan in Kurokawa Onsen. Natural hot spring (sulphur spring, per the inn)

Tattoo rule
What the official page says
当館はタトゥーを制限しておりません。 今までにタトゥーのある方が沢山お越しいただいてますが、特にトラブルはござません。
We do not restrict tattoos. Many guests with tattoos have stayed with us, with no particular trouble.
The official FAQ, in Japanese and English, says the inn does not restrict tattoos.
Kurokawa Onsen is known for its bath-hopping pass (nyuto tegata).
Where it is
Kyushu region, Kumamoto prefecture. About 89 km in a straight line from Fukuoka.
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Rules can change. Check the official page on the day you go. Any paid link is labelled next to the button. We have not visited this bath ourselves.