Mihoshi Special
The Mihoshi Special may have originally been meant to be a part of the Tenchi Muyo! Ryo-Ohki canon, serves as a partial introduction of Kiyone Makibi for Tenchi Universe, and of Pretty Sammy, but it ultimately stands on its own. In the story, after a day of lazing around at home, Mihoshi defends her competence as a GP officer by recalling a story of her and Kiyone's battle with a notorious space pirate. For convenience sake, she casts Ryoko as the pirate, Washu as the mad scientist at the center of the pirate's actions and the rest of the household as other characters in her story. As her story becomes more and more ridiculous, she even gives Sasami's Pretty Sammy identity a heroic role.
Ai Tenchi Muyo!
In a new continuity that seems largely related to Tenchi Universe, with small differences, an experiment of Washu's has caused a big mess by displacing a couple Juraian travelers in space and time. Two versions of Mihoshi show up in the series, one shows up hundreds of years ago in Japan, This version of Mihoshi seems a bit more together than her later self. She's a Detective First Class on a mission to capture the wanted Space Pirate Ryoko (before either knew Tenchi) and ends up disappearing in a dimensional rift with Ryoko. What happened directly after is not revealed.
In the present day, airheaded Mihoshi is helping a group excavate the Juraian's spaceship, though she seems to believe she's on a simple treasure hunt. She says she is supposed to be on a mission to capture Washu, but thought a treasure hunt was more fun.
Kiyone Makibi does not appear in this series. Instead Mihoshi works with Ukan Kurihara, a superior officer she has little contact with.
Tenchi Muyo! Daughter of Darkness
The movie, Daughter of Darkness, was actually based off the 12th out of 13 novels written by Naoko Hasegawa, a scriptwriter for the first season of Tenchi Muyo! Ryo-Ohki. Mihoshi has a strictly supporting role in the film and doesn't contribute much to its short length.
Hasegawa novels
This film is the only contact most non-Japanese fans have had with Hasegawa branch continuity. The first book of the series, The Hexagon of Love, was produced with the cooperation of Masaki Kajishima and briefly considered original canon. The second and third novels retell the story of the first season of Tenchi Muyo! Ryo-Ohki. but afterwards branch off into their own direction which is sort of a combination of both the OVA and Universe canons. None of the novels have been officially translated outside of Japanese, so little is known about the character development within them.