Barbatos is the Anemo Archon and one of the original 7 archons that won the Archon War (a war from 2000 years ago between gods attempting to claim 7 divine thrones). He is known as the God of Freedom and is also known as an absentee god, allowing Mondstadt to govern itself.
At the beginning of the archon war, Barbatos was merely an elemental wind spirit and had become friends with a bard. This bard lived under the control of the god of storms, Decarabian, who concealed his city within a harsh wind storm that no one could go through after the other god in the land, Boreas, declared war on him.
Eventually this bard led a revolution with many others. Decarabian was killed, however the bard lost his life in turn. Barbatos had taken on the appearance of the bard and quickly claimed one of the divine thrones, as, Boreas, had stepped down, seeing that he could not love this land and its people like Barbatos did.
Barbatos would then use his powers as a god to level the mountainous land of Mondstadt and carried away the snow that once covered it, eventually finding the main city of Mondstadt. Barbatos would then fall into a deep sleep and reawaken in times of trouble, such as the Cataclysm and the dictoral rule of the Lawrence Clan.
At the beginning of the main story, the traveler meets a bard named Venti, who is seen comforting a dragon, Divalin, one of the Four Winds (4 entites that have been entrusted with the duty of protecting Mondstadt). It is evident throughout the story that Venti is Barbatos himself.
Interestingly, Venti, during the Mondstadt Archon Quest, states that he is currently the weakest of all the seven, claiming that an Archon's power is derived from how much control they have over their respective nation. However, in the Sumeru Archon Quest, Archon Buer disputes this,
stating that an Archon's power actually comes from how much they are worshipped. Venti has a church and statue dedicated to him and is one of the few archons who has worshippers all over Teyvat. So while he may not be the strongest, he is far from being the weakest.
Istaroth is known as the God of Time and is one of the four shining shades of the Primordial One (also known as the Heavenly Principles). She goes by many names, such as "the thousand winds", Tokoyo Ookami, and Kairos, and many more.
She is worshipped widely in Mondstadt and Ekanomiya, although that worship has faded by the time of the game's story. In fact, most people no longer know that she exists, with her legacy remaining through the phrase "winds of time".
As for Ekanomiya, they continued to worship her after their island sunk below the surface, but that worship began to fade with the death of their god, Orobashi.
She was worshipped with Barbatos at the same time for a lot of Mondstadt's history and they likely had a temple built to worship both of them. This temple is known the "thousand winds temple", albeit in ruins now.
In fact, with the Lawrence's rule over Mondstadt, they likely contributed to the loss of information on Istaroth, destroying and hiding artifacts to hide the evidence of their misdeeds.
After the fall of their rule however, a great fire would break out, and reduce Mondstadt's library (formerly the largest library in all of teyvat) to a sixth of its size.
She has made several appearances throughout the side stories of Genshin Impact, from the mysterious island quests in Mondstadt to her helping the former Archon Baal in Ei's story quest.
Istaroth is unique through her constant presence in the world despite having next to no information on her, and many fans speculate that she has done more to alter several factors within the history of Teyvat.
As noted in the section, they are connected through Istaroth's many titles and names as well as their combined rule over the nation. However, there is nothing in the game that explicitely states their direct connection.
Many people have theorized that Venti is a piece of Istaroth, taken from Venti's description of himself, he was just a "single thread of the thousand winds that roared through the northern lands". Since Istaroth is known as "the thousand winds"
many people have taken this to be Venti formerly being a piece of the shining shade.
Of course, there are several theories around this, such as Venti being istaroth himself and how Istaroth will be incredibly important to either the main story or Venti's story.
There are many speculations that Istaroth will make an appearance in some form in one of Venti's story quest chapters. Which is interesting as most older archons (game release-wise) have already gotten a 2nd chapter in their story quests. Venti is seen to be the most mysterious
out of all the archons in terms of what he knows as well his connections to other gods. As such, it'll be interesting to see what direction Genshin Impact will take his story.
Fun fact: Venti's connection to Istaroth was teased back in version 1.2 through Albedo's character demo.
At 1:23, the first shot displays text that reads time, and in the very next shot, there is a cecillia which is a flower closely related to Venti (as seen in his hat)
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