Path

Paths (道) are channels that connect all Titans and Subjects of Ymir together.

Overview
The idea of paths came about when people noticed that the Nine Titans are sent to a random baby among the Subjects of Ymir, regardless of distance or blood relations, if they die without being eaten. This implies that the Subjects of Ymir are all connected. What connects them are paths.

Paths carry many things, such as memories and wills from other Subjects of Ymir, flesh and bones that make up a Titan's body, commands from a Titan's scream, and the power of the Titans.

Family seems to be relevant for paths: Memories are transferred far easier between between family members. Karl Fritz's vow renouncing war is only passed down to his own family. The Ackerman clan, who are also connected to each other by a path, gain the battle experiences of all past Ackermans when they awaken.

Paths transcend physical space, and they appear to transcend time as well, for Eren Kruger had "memories" of Mikasa Ackerman and Armin Arlert before they were even born. Grisha Yeager likens paths to magic.

Alternate World
Paths can only be seen in an alternate world, covered in white sand. Paths are shown as streaks of light in the sky, similar in appearance to an aurora. Paths are invisible in the physical world.

The world appears to be a separate reality. Ymir entered it after she gained the power of the Titans. Zeke Yeager entered it by having a near-death experience. Time is irrelevant there: Zeke was only there for a short moment, but to him, it felt like years.

The world is inhabited by a young girl carrying a pale of water.

Coordinate
When paths cross, the point is known as a "coordinate" (座標). The Founding Titan, which is sometimes called "the Coordinate" by the Warriors, is the point where all paths of the Subjects of Ymir and Titans cross. It has the ability to alter the memories of Subjects of Ymir, change the body composition of Subjects of Ymir, transform Subjects of Ymir into Titans (even Colossus Titans), and command all other Titans. According to Tom Xaver, the Subjects of Ymir are like a part of the Founding Titan's body, no matter where or how many.

If Zeke Yeager's spinal fluid enters the body of a Subject of Ymir, that Subject of Ymir becomes connected to Zeke's Beast Titan by a path and is marked with a coordinate. Zeke's Beast Titan can act like a miniature Founding Titan to those who are connected to him. He can send the power of the Titans to transform the affected Subjects of Ymir into Titans, and he can give those Titans commands.

Only those with royal blood have access to a coordinate's power. Only those with royal blood can use the Founding Titan's power, and the reason why Zeke can make his Beast a coordinate like the Founder is because of his royal blood.

History
According to Eren Yeager and Kruger, one of their predecessors saw paths.

During the year 845, after Ymir devoured Marcel Galliard and gained the power of the Titans, she awoke in the desert world and saw paths in the sky, calling them "freedom" spread out before her. She vowed to live a life without regrets afterward.

Unnamed final arc
As Zeke Yeager is dying from the wounds inflicted by a Thunder Spear, he has a vision of a young girl in a world full of sand. A Titan appears out of the ground and opens its abdomen to put Zeke's near-dead body inside. Zeke awakes in the world to the girl rebuilding his body using the surrounding earth, and sees paths in the sky above. Zeke comes out of the Titan fully recovered.