Saw this post yesterday but wanted some time to gather my thoughts a bit.

First, let me second the recs for Bodrov's Mongol (though idk about accuracy) and manga Kingdom (actually surprisingly well researched, but obviously exaggerated presentation due to the medium and takes a lot of liberties)

Second, "medieval" is a rather broad term coming from a layperson. There are very distinct periods of Ye Olde Asia that can't be encapsulated by that word. I'd argue that what most people in Asia visualize of "old China" is probably either Tang dynasty (very accessible period due to various reasons) or Romance of the Three Kingdoms (very fictionalized take on history, ymmv GREATLY on accuracy), although from Western perspective I think the usual visualization is Qing. Similarly, "old Japan" is mostly old Edo or Sengoku Jidai. I'm not as familiar with Korean history so I'll refrain from commenting on that.

My personal area of interest leans toward pre-imperial China, which you're pretty much just never gonna get an accurate take on in pop culture due to lack of archaeological evidence prior to relatively recently. (hint: if you see copious amounts of horseback riding the accuracy level automatically plummets.) However, I want to give a nod to the movie Confucius from 2010 -- boring/meh movie but the research and set/costume detail is probably as solid as you're gonna get.

I'd also highly recommend The Last Supper from 2012 once you've familiarized yourself with the basic Chu-Han storyline. The Last Supper is a subversive leftist take on the traditional narrative. There's some weird CG in the movie but otherwise I'd say the accuracy leans toward the higher end.

As for Japanese history, it's actually very easy to get good visuals for this via NHK taigas. Although accuracy may vary depending on era, these dramas used to be Very Prestigious and taken pretty damn seriously so you can usually trust the accuracy (relative to what's available when the drama in question was filmed). I think one of the more visually distinct eras of JP history is actually the Heian era (look for Tale of Genji adaptations)... but as I said above you'll usually find other periods more popular.

For a quick overview on what's popularly dramatized and periods with a general unified aesthetic...

Chinese history:

1. Early imperial

• The Qin Emperor has historically been very controversial so it's a mixed bag on what you'll get about him. It's usually about the assassination attempts on him. Just read Kingdom tbh

• Chu-Han contention. Again, I wouldn't pick Last Supper as my first exposure to the story, but once you have basic familiarity I think it's my favorite take

• Three Kingdoms... uh where to even start with this. Just be aware that RoTK is historical historical fiction lol, and that's reflected in many adaptations (especially the older ones).

2. Tang dynasty. Sorry, no specific recs but if it's colorful it's probably Tang, lol. Most pseudo-Chinese fantasy from Japan is also vaguely Tang (Twelve Kingdoms, Kusuriya no Hitorigoto, Fushigi Yuugi, among others -- rec the first two but not the third). Note this is a huge time jump from the previous category. You'll occasionally get stuff between the Three Kingdoms era and the Tang dynasty but it's not popular, lol.

3. "post-Tang". My familiarity with later periods is pretty shit haha, but I can tell you that a lot of classic wuxia (at least Jin Yong/Louis Cha's wuxia, which is what a lot of people will automatically visualize when thinking "wuxia") is post-Tang. Condor Trilogy is mostly Song dynasty with the third book being about the transition from Yuan to Ming. Not sure I'd watch wuxia for historical accuracy lol but I do think Jin Yong is a cultural touchstone that you want at least some familiarity with to catch most of the references to his works that pop up in games and other media.

4. Qing into revolutionary era... ummm also not gonna rec as I just don't watch a lot of the more modern periods

Japanese history (mine is a bit spotty so I may miss stuff):

1. "ancient"/Yamatotakeru/Himiko era: all super fictionalized. For the most acceessible down-to-earth depiction I've seen you'll probably have to dig through the Professor Munakata series for relevant cases (Munakata suffers from terrible pop history sometimes but at least the visuals are convincing)

2. Heian era: again, just look up Genji adaptations. Abe no Seimei is also popular but bound to lean too fantasy due to the subject matter.

3. Heike Monogatari. Surprisingly not AS popular in modern pop culture despite its historical popularity... there's definitely a taiga drama (or three) on it though.

4. Sengoku Jidai. No recs because it's too damn popular, Japan's equivalent of RoTK. You want THE essence of pop culture old Japan, you look here.

5. Late Edo/Meiji. Also popular, also tons of taiga dramas... Shinsengumi stuff is really popular (boo cops), Sakamoto Ryoma is popular and a lot more fun as a historical figure, Gintama is a completely unhinged science fiction-tinged parody... hard to rec because like the Sengoku stuff it's so popular most of it is super fictionalized at this point.

6. early 20th century/Taisho. Maybe starting to get more popular due to Golden Kamuy (highly rec, well researched Ainu culture featuring an insane cast + pop culture jokes) and Kimetsu no Yaiba (this is fantasy, the historical setting is purely a vibe) in more recent years. Saka no Ue no Kumo (Clouds Above the Hill) is said to be the definitive historical fiction take but the English translation of the novel is a bit dry and I haven't watched the drama adaptation and either way this isn't really what you're thinking when you think "medieval Japan" so never mind I got sidetracked lol

... Uhhhh apologies for formatting