Because our beloved Lore and Art teams are absolute lunatics it looks like they've created an entirely new script to use in Doma; a simple substitution cipher for hiragana similar to how Eorzean works for the Roman alphabet.
ETA - JP fandom, thankfully, has already got this mostly worked out.
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I don't think there's enough text in the game yet to get a full syllabary but this is what I've got so far;
(Note - I'm using Kunrei-shiki romanisation for this instead of Hepburn because it makes it easier to follow how many kana you should expect to see.)
KOUBAI GOTEN (紅梅御殿) Shirogane Housing Appartments
An apartment sign is a weird sort of Rosetta Stone but this was really helpful because it had こ (ko) and its voiced version ご (go), which in turn lets you know what the diacritic marker looks like, which made it easy to identify ば (ba).
From there it's easy to figure out the rest of the script.
BOUKAISEN (望海泉) Kugane Hotsprings
This could have also been Boukairou (望海楼), the inn attached to the hot springs, but we've already seen the kana for ん (n) in the sign for Koubai Goten so we know it's Boukaisen.
. . . Aaaaand that's all I've managed to get for now. Hopefully the JP side of the fandom is also already working on it and is a lot further a long?
(Many thanks to Tenchiko Uitsurugi and Douse Sekkaku of Leviathan for helping me find text examples and putting up with my ranting, this blog for some of the Kanji readings and this reddit post for setting me off on this madness.)