Kyushu Kumamoto
Kumamoto-jo Sauna Shiro no Yu
熊本城サウナ 城の湯
Name in Japanese. Look for this on the sign, or show it to a taxi driver.
Allowed with conditionsStated by the facility
Sauna and bath facility. Hot spring (spring details not checked)

Tattoo rule
Allowed with conditionsStated by the facility
What the official page says
タトゥー入れ墨がありますが入店できますか? 22時以降は入店可能です。(誓約書必須)
In short: Guests with tattoos can enter after 10 pm and must sign a pledge. At other times a cover sticker is required.
Where it is
Kyushu region, Kumamoto prefecture. About 91 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.