Rakugo Tennyo Oyui

Yui Tsukishima is an Ordinary High School Student who aspires to be a Rakugoka. Rakugo is a Japanese comedic storytelling art, relying only on the voice and a fan. She attends school with the shy Tae Yanaka, swordswoman Ryo Sengoku and her chief rival, Miyabi Asukayama. After pestering Master Utamaru, a professional rakugoka, enough to be taken on as his apprentice, she finds that she and her fellow classmates, along with child prodigy Suzu Koishikawa and inventor Akira Naitou, are about to be taken to the past.

The Tennyo (celestial maidens), protectors of Edo, have been defeated. Unable to fight off a recent invasion of demons, the jewels within them travel to the present and bring in their reincarnations - Yui and her friends. They now must adapt to a new time period, discover the powers within themselves, and save the city from the invading army.

Rakugo Tennyo Oyui is an Anime First series that aired in the winter of 2006.

Tropes used in Rakugo Tennyo Oyui include:

Accidental Kiss -- Yui and Miyabi in episode 11. Neither of them is happy about it.
Alpha Bitch -- Miyabi. She doesn't let up even when everyone's in the Edo period.
Anguished Declaration of Love -- Gennai does this to Akira. As it turns out, this boosts Suzu's healing power enough to revive her, and helps Akira revive herself.
The Archer -- Miyabi's weapon of choice.
As Himself -- Utamaru, the rakugoka who Yui begs to be apprenticed to, is both real and voiced by himself.
Barrier Maiden -- All six Tennyo's jewels reinforce the barrier around Edo. Though for a barrier maiden, they're pretty active in the field.
Barrier Warrior -- Suzu's power by itself is weak, but she can reinforce the others long enough.
Cheerful Child -- Suzu
Combat Tentacles -- The giant cyclops...thing summoned in episode 8.
Convection, Schmonvection -- Yui was protected by a magical barrier when Enchou's first home was on fire, but Tae wasn't.
Cute Little Fang -- Yui
Dangerous Forbidden Technique -- Yui's otoshi-banashi technique. As it's a fan slash from above, the stronger the enemy, the greater the height she needs. In the case of the black dragon, she's on the border between the atmosphere and space, with the risk of death if she screws it up.
Debut Queue -- Though everyone is in Edo by episode 3, it takes one episode at a time for their powers to awaken to their maximum.
Diegetic Switch -- One of the closing moments of the show has Hiromi Kashima, who sings the ED, appear on TV, and then cuts to the ED.
Five-Man Band
- The Hero -- Yui (Rakugo Tennyo)
- The Lancer -- Miyabi (Kotodama Tennyo)
- The Smart Guy -- Suzu (Keiryaku Tennyo)/Akira (Karakuri Tennyo)
- The Big Guy -- Ryo (Kenkaku Tennyo)
- The Chick -- Tae (Kagura Tennyo)
Fish Out of Temporal Water -- From the year 2005 to Keio 3, which puts them in 1867 on the Gregorian calendar. They all adjust surprisingly well.
Genki Girl -- Yui, also The Ditz.
Healing Hands -- Suzu's Tennyo abilities include these.
Historical Domain Character -- Enchou Sanyuutei, a real historical Rakugoka. Gennai Hiraga, a famous inventor (and Akira's love interest.) Toshizou Hijikata, commander of the Shinsengumi. All fairly major characters.
Hot Chick with a Sword -- Ryo
Hot Scientist -- Akira
Idiot Hair -- Yui.
Intergenerational Friendship -- Akira's closest companion is Suzu, the 12-year old genius.
In the Name of the Moon -- Each of the girls has a rather lengthy phrase when they transform.
Jidai Geki
Katanas Are Just Better -- Ryo
Legacy Character -- Even if the jewels forcibly teleport their users to the past.
Magical Girl
Meganekko -- Tae.
Mind Control Eyes -- Tae, for her duration of the wedding and when she and Ukyo are turned into the black dragon.
Monster of the Week
No, Mr. Bond, I Expect You to Dine -- Sakyou treats Yui and Enchou to a meal before their arranged final battle.
Ojou -- Miyabi
Otaku -- Suzu in particular. She's an Edo fan, and is super excited about actually being there.
Otaku O'Clock
Paper Fan of Doom -- Enchou, and also Yui's weapon once she awakens her powers.
Power Gives You Wings -- All Tennyo, sans Akira and Ryo, have a pair of glowing wings on their back.
Puppy Love -- Suzu and Ponta. Though Ponta's a tanuki, so he may be somewhat older than her, but their mental ages are about equal. Also an Interspecies Romance.
She Cleans Up Nicely -- Everyone is excited to see the tomboyish Ryo dressed up in a kimono. It does look good on her. Hijikata especially thinks so.
She Is Not My Girlfriend -- Though Ponta says so, his grandfather still thinks Suzu is. Suzu also denies it.
Shrinking Violet -- Tae
The Strategist -- Suzu's power is noted to be this.
Teen Genius -- Suzu
This Is My Boomstick -- When they're stuck without a kimono for blending in, Suzu and Akira impress a pair of local shopkeepers with their modern cellphones. Even without reception, the camera is enough to wow them.
This and That -- As Miyabi had yet to do this and that, her first order of business is to spy on Yui and figure out what "this and that" is.
Title Drop -- Yui's Role Called title is dropped at the end of episode four when she awakens her true power as a Tennyo.
Transformation Sequence
Transforming Mecha -- A karakuri variation. It can go from vaguely humanoid to a tank, but it's powered by Akira's own jewel energy, and she hadn't awakened at the time. It's also rebuilt into a time machine(!) in the final episode.
True Love's Kiss -- Gennai wakes up Akira in this fashion when she's trapped in a Lotus Eater Machine.
Unholy Matrimony -- Ukyo is to be wed to Tae in hopes of making her evil. Tae resists, though.
What Do You Mean It's Not for Little Girls?
Where Are They Now? Epilogue -- As narrated by Yui. Notably, Tae becomes the Student Council President.
You Have Outlived Your Usefulness -- When the wedding doesn't go through, Sakyou kills his younger brother on the spot. Since it makes his older brother happy, Ukyo doesn't resist.