I've seen this idea in various forms in these forums, but here's a spin that I think fits in to the world of Sea of Thieves in both aesthetic and theme, while not being too HUD/mechanic heavy or confusing.

The Storyteller's Society

Like Mike Chapman has said in the various videos, each faction in Sea of Thieves has their own intentions. The Merchant's Alliance wants to control trade. The Gold Hoarders want to amass wealth. What if there was a faction who has come to the Sea of Thieves in search of these stories of pirates and merchants, and wants to share them with the world?

Enter the Storyteller's Society. As mentioned, the Storyteller's Society wants to learn about the deeds of these pirates, and write books and weave tales about these emergent pirate legends. But any pirate can come up to a storyteller and lie about their deeds. These writers and poets want something better- some proof that the pirate is as great at finding gold, or as great as sinking ships as they claim to be.

Mechanics

Storyteller's Society reputation is gained in a slightly different way than other factions. Rather than having set "voyages" that you embark on that have a concrete end, reputation (and gold!) is gained in this faction by playing the game as you normally would. However, they provide you with a tool, purchasable for a small fee (same as a voyage): The Storyteller's Flag.

The Storyteller's Flag

In the "voyage select" screen, in lieu of voyages, you are presented with three flags, each costing the same price (50g?)

The Pirate's Flag (Classic Black jolly roger flag)

The Voyager's Flag (Blue flag w/ picture of spyglass)

The Cartographer's Flag (Green flag with world icon on it)

Upon purchase of the flag, you are allowed to pick up the flag from the NPC, like you can with Pig Crates. It, like every object in Sea of Thieves, is a physical object that can be carried, and of course, stolen. In order to begin accruing progress in this faction, the flag needs to be carried to the top of the crows nest, where it can then be equipped to the top.

While a flag is being flown, certain actions you and your crew take will begin to add Proof to the flag. If a player sinks your ship and captures your flag, they can turn it in to the Storytellers, but cannot equip it and continue to gain Proof- only the flag that was purchased by your crew can be equipped and unequipped, to avoid people switching flags out over and over again between outpost trips. A flag can always be unequipped from the crow's nest, which would then return it to it's default (currently White) image.

Gaining Proof

Dependent on the flag that was selected, certain deeds begin to add "Proof" the the flag:

The Pirate's Flag

Proof is gained in this flag by doing PvP and Hostile actions.

A small amount of proof would be gained for every enemy player defeated.

A larger amount of proof for every ship sunk.

The Voyager's Flag

Proof is gained in this flag for every "voyage" action accomplished.

A small amount of proof would be gained for intermediate voyage actions (getting the "THUNK" of unearthing a Gold Hoarder chest, capturing the required animal for Merchant Alliance, spawning the Captain in a Order of Souls quest).

A large amount of proof is added upon the successful completion of a voyage.

The Cartographer's Flag

Proof is gained in this flag for traveling the world, and finding uncharted and unknown things.

A small amount of proof is added for setting foot on every unique island.

A large amount of proof is added for entering shipwrecks.

Leveling up and Turning in Flags

As long as a certain flag is equipped to your crow's nest, these actions continue to add Proof to your currently equipped flag. As the flag gains more and more Proof, the visual quality will get higher and higher, as well as the "name" of the flag.

Flag Levels

Similar to "Rare" Tea or "Exquisite" Silk, the name of the flag would imply it's value upon turn-in.

Fresh Flag - 0g

Experienced Flag - Similar to Seafarer's Chest

Veteran Flag - Similar to Marauder's Chest

Legendary Flag: Similar to Captain's Chest

Example Storyteller Gameplay Loop

Crew A logs on and decides that today, they want to do some fun voyages. They go to the Storyteller's Society and buy the "Voyager's Flag". They pick up the "Fresh Voyager Flag" item, and equip it on their crow's nest. Then, they decide to pick up some Gold Hoarder voyages and go on their way.

As they play, their flag begins to grow in quality. The "Fresh Voyager Flag" becomes an "Experienced Voyager Flag" after two voyages, and they contemplate turning it in right then and there since they know those usually earn ~200g. They decide to keep going, and after 4 more voyages their "Experienced Voyager Flag" has become a "Veteran Voyager Flag". Now their flag has "upgraded" from the spyglass to two crossed spyglasses, indicating its rank. They decide to try a few more voyages to level it up all the way to "Legendary Voyager Flag".

Meanwhile- Pirate Crew B just logged on and decide they want to do some PvP. They go to the Storyteller's Society and pick up a Pirate Flag. They equip it to their mast, and begin searching for players. While sailing between Plunder Outpost and Ancient Spire, they spot a ship parked on Cutlass Cay. As they get near, they see the flag and - what do you know? Two Crossed Spyglasses. The game is on!

Crew A sees Crew B on their way, and they can see the Pirate Flag flying. It's not very experienced, looks probably like it's just a Fresh Flag, so they know they either just started, or had just turned one in. Either way, they aren't giving up their treasure (and flag!) without a fight!

During the fight, Crew B manages to kill 2 of Crew A's players, upgrading their flag to an "Experienced Pirate Flag." In the mayhem, however, they don't realize Crew A had been saving a Gunpowder Barrel for just an occasion. The captain of Crew A sets the barrel off and- BOOM! Crew B's ship begins to sink and they forgot to keep somebody on board to repair. Crew B's ship is done.

While searching the wreckage of Crew B's ship, Crew A finds their old Experienced Pirate Flag. They pick it up, and put it in the captain's room. Even if they wanted to equip it they couldn't, since it was tied to the original boat. Either way, they decide they've had enough excitement for one day. They finish up their current voyage, and head to the outpost to turn in their Chests, their (now Legendary!) Voyager Flag, and the Experienced Pirate Flag they got from their would-be plunderers.

Benefits of this Faction

Increased Risk leading to Increased Reward

Besides time being wasted, there's currently no real downside to turning in chests and reward items after every mission. This leads to many ships with mostly empty hulls sailing the seas.

The Proof gaining aspect and "leveling up" of these flags would lead to ships having the option and incentive to staying out longer between outpost visits, which increases both the feeling of relief once the flags and chests are turned in, as well as the feeling of loss if their gamble of staying out too long does not pay off.

Displaying of Intent

If you see a ship with a Pirate Flag headed your way, you can probably be sure they're looking for a fight. This is a good and bad thing for both parties- a ship that doesn't want to fight may decide that they want to immediately escape, rather than stick around to see what the ship does. That being said, a ship that is also looking for a fight will be happy to oblige the approaching ship, and both ships can be sure that rather than a game of cat and mouse, this encounter will lead to a fun, engaging ship battle with a reward going to the victor, even if both ships had no chests on board.

Like anything in the Sea of Thieves, the grounds for deceit are there. There's nothing stopping a would-be plundering crew from flying a non-pirate flag to try and deceive others. That's a totally valid way to play as well! It just means that they will not be receiving the additional gold and reputation rewards for flying the Pirate Flag, so there is an incentive to being honest with your true intentions under this system.

Conclusion

I think the introduction of the Storyteller's Society would be an additive and complementary change to the Sea of Thieves voyage system, with many game-play benefits, as well as allowing as fitting into the lore and aesthetic of these explorers in this new land.

Thanks for reading my pretty long post, and let me know what you think! Definitely up-vote if you think something like this would be a cool addition to the game to try and get visibility!