A classical depiction of an astral leaving after death.
The Astral Pattern, often simply called an Astral, is the closest equivalent to a soul known in the universe of Tenchi Muyo! Ryo-Ohki. While somewhat unique in concept, as used in Tenchi Muyo! Ryo-Ohki and surrounding series, It is largely associated with the larger theoretical concepts of the Astral Body, Subtle Body, Astral Plane, and the Akashic Record, adopted by many of Earth's philosophies, particularly in the East.
Background
While little mentioned in the anime, Astrals play a large part in the background of the True Tenchi novels, particularly the Tenchi Muyo! GXP novels.
As explained to Seina Yamada by his teammates Kenneth and Rajau, there exists something called the Astral Sea. When a life form or AI achieves sentience, a form or 'pattern' takes shape within the Sea, and contains all the individuals knowledge, memories, experience, everything that makes them an individual. An Astral is usually completely unique though. A newly born entity can use a previous astral as its core, but the result will still be a new astral that incorporates the previous one but is, in essence, anew astral. Cloning a person, or copying the program of an AI with an astral will not copy the astral.
Once an individual passes on, the Astral Pattern will return to the Astral Sea, bringing with it all the memories of the person it belonged to. Because of this, when a person ends up inheriting a significant portion of a previous astral, that individuals will very rarely experience residual memories of a long dead person.
There is, however, one way to copy an astral that Kenneth and Rajau would not know about. It is mentioned first in True Tenchi Muyo! Volume 1: Jurai novel that creating an astral copy is only possible using the extreme power of a 2nd Generation Royal Tree or above. These are not identical to an astral that has returned to the Astral Sea but a copy of an individual's astral created from a particular period of time. The Jurai novel describes how, Masaki Jurai created an astral copy of herself and stored it on Earth's moon, where it survived well into the future after the original Masaki passed away. Although it is not explicitly stated, It also seems that D was able to make an astral copy of herself that reflected her astral before she became "Dee Sanada" and this copy survived a billion years into the future, becoming the version of D that became a cabbit and a companion of Seina. After all, "Dee Sanada" was a very different, vivacious and engaging young girl after the dimensions were combined. Oddly though, the future D does seem to have the memories of Dee after the dimensions were combined.
In rare circumstances, so rare that they are considered a myth, individuals can have what appears to be an Astral Pattern that uses so much of the core of a deceased individual's core that it makes the memories of the previous person even more likely to be remembered, under the proper stimulus. An example being Seina Yamada and Kazuki Yotsuga. It seems that the effect of the lives of Kazuki and his companions had such an effect on the Astral Sea that aspects of their astral patterns were widely spread throughout history and all over the galaxy. Because of the relationship between the original astrals, those that contain the remnants of these repeated astral patterns find themselves attracted to each other, particularly between the astrals of Kazuki and his harem of girls.
See Also: Seina's Wives
With the proper technology, an individual's Astral may be retained after death, and even transferred into a new body as the core of a new astral. Such was the case of Widow and Fuku. Washu wanted to preserve the extreme dedication to Seina Yamada that Widow had developed and so she used Widow's Astral as the core of Fuku's astral. It is mentioned that Washu deliberately erased Widow's memories from Fuku, but she seemingly retained the extreme devotion Widow had developed for Seina.