Got around trying Tanuki and Moriya Shrine. Both offered entertaining games, although Moriya felt a little bit too powerful.
Tanuki's UA was fun to use and offered a nice bonus for early game economy and also gives nice intel for early wars. Earlier tech stealing wasn't nearly as huge deal as I was worried beforehand, although I played on Emperor; don't know how much it affects the game on higher difficulties. On Emperor it gave some cheaper techs but wasn't really gamebreaking.
+2 global happiness by Classical and +4 by Medieval might be a little too much when combined with the gold those spies also offer, but it is still not really gamebreaking bonus. The nice thing about this UA is that it scales on it's own to later stages to the game when bigger cities give more money for spies. Overall a really nice UA.
Their unique scout is very handy and makes early scouting much easier. It is also nice that when you are done with scouting, you can also get a little extra gold from their unique action before deleteing them. Really, a modest but very convenient unit.
Their bank replacement is also pretty nifty with a small boost to happiness and extra income from trade routes.
Overall a nice civ with many small, convenient bonuses which when combined feel satisfying to use
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As a side note, though: this civ seems to mess up unit command panel when using Enhanced User Interface:
Not a huge deal, just letting you know.
Moriya Shrine, on the other hand, has very real potential to be pretty overpowered, provided you manage to get your Faith production high enough.
Their UA is pretty interesting. Getting Faith for discovering technologies is really strong, perhaps a little bit too so. It honestly felt like a better version of Celts, since both get faith easier in early game, Moriya just gets more Faith for less effort than Celts. This trivializes the early religion raze.
Getting 1 science for every 2 faith is also really interesting. It can get a little bit out of hand in early game if you manage to get good faith pantheon (I manged to get +2 Faith from Gems and Pearls with 4 Gems near my capital) but in most situations is should give reasonable bonus. In later game it might get out of hand again, depending how much Faith you can produce. I got to +120 faith per turn by Industrial Era, and my tech progression honestly felt to be too fast compared to normal.
I really like the fact that you get Science and Faith from mountains that are inside cities' borders. This makes those tiles less useless and might encourage players to find cities in unusual places.
Overall this UA is interesting and in most situation it's probably reasonable, but it has real potential to get out of control.
Yamato Swordman has interesting promotion, but it also feels to be a little bit too good. If I understood correctly, it maxes out at +6 strenght at 24 Faith per turn (not that hard to get). +6 strenght feels to be too much, ecspecially since this promotion also transfers to upgraded units. This pretty much lets them dominate all melee combat after Iron Working, provided that they can keep them alive.
I honestly think that this bonus should be smaller, maybe max +3 or 4. Right now it just makes Moriya Shrine too good at combat.
Taisha is very intriguing, since it doesn't give any faith on its own, just from religion followers in the city. I don't if this was intented since their Workshop page says that it should get also Temple's normal bonuses, but right now it has very good mechanic: it is totally useless on its own, but has potential being much better than the normal Temple if Moriya's religion is spread efficently enough. This makes it interesting and also gives a small risk element. Overall this makes Taisha more intersting to use.
I don't know if this is supposed to be included with this civ since there is no mention of it on the Workshop, but Moriya also has unique Grand Temple (named Moriya Shrine). Honestly it's Science bonuses (+4 Science, +25% Science) feel a little bit too good wen combined with Moriya's other science boosts and also causes them to have more unique components than other civs. This honestly helps to make them too strong when compared to other civs.
This unique Grand Temple also apparently boosts buildings called Onbashira. I have no real idea what they are since I couldn't make them, but based on Civilopedia searching I guess they have something to do with Events and Decisions.
All in all, both of these civs have very intersting ideas in them. Tanuki has many small bonuses that together make a very interesting and to use package. Moriya Shrine also has many interesting ideas: making a civ stronger based on its Faith production is an unique and fun mechanic to use. Right now, however, their bonuses seem to be too big, ecspecially if they manage to get big amounts of Faith production. They probably would become more enjoyable if their bonuses where cut somewhat.