This trope is about Japanese works that portray Christian nuns as being similar to Miko (shrine maidens, shamans, spirit mediums).
Few Japanese people today are overtly religious, and even fewer are Christians (less than two percent of the population). So Christianity, particularly the Roman Catholic type, tends to get used as an exotic religion, especially if you can mix in mythology and folklore. Some writers are fannish enough to do research, especially if a story takes place in a Medieval European Fantasy. But most have only a basic understanding of the religion, and few writers have a realistic idea of what a nun's life is like.
For aesthetic purposes, this depiction always resembles Roman Catholicism; Orthodox and Protestant trappings are never seen. Christian or Pseudo-Christian monks are conspicuous by their absence, while there will be a noticeable number of pretty nuns. Aesthetics aside, expect a substitution of personalities for the eastern equivalents.
Instead of the Western role of the nun as a cloistered celibate religious figure, the nun in this role will often be a nun as a part-time after-school job, or only until she marries and not take a vow of celibacy, poverty or obedience. Her role will be more focused on divination than the traditional nun. Her spiritual powers will also resemble the miko—being able to see demons or produce a shield against them.
If the show goes in for high fantasy, the association with miko follows suit: she is just as likely to fight demons, use weapons (especially a high-calibre and potentially magically-enhanced gun) and have suggestive relationships.
Slightly more excusable in the Crystal Dragon Jesus religions, which is perhaps its western equivalent. Often meshes with the Church Militant. In the rare times you have both a nun and a miko, the nun is usually depicted as more patient and demure. See also Hijacked by Jesus and All Monks Know Kung-Fu.
Compare Naughty Nuns, Nun Too Holy. Ultimately also a subtrope of Creator Provincialism, but it's prevalent enough to warrant its own page. This trope is one of the more common causes of Anime Catholicism- if the inaccuracy is about something other than nuns, please place it there.
In the anime canon, it's explained that the Magdalene Order is actually comprised of many different religious branches working together for the common goal of combating demons. Rosette herself is a teenaged girl who despite her religious occupation doesn't have a religious vocation. (She joined the order because they were the only people who could give her the training and resources she needed to save her brother.) She even seems to think the handsome Priest will notice her charms one day.
Fate/Zero has a male example with Risei and Kirei, two Catholic priests, the first the father of the second whose wife recently passed away.
This kind of situation is actually possible in real life, but is incredibly uncommon.
Seira from Kaitou Saint Tail is a part-time "nun-in-training" after school, and wears a white nun's habit — all analagous to a miko's role, but antithetical to nundom. Tachikawa Megumi admits in volume notes that she didn't do her research after being corrected in a fan letter from an actual nun.
Downplayed and mostly averted in Hellsing. Yumiko "Yumie" Takagi behaves very much like a Catholic nun and her only noticeably Japanese traits are her name and penchant for wielding katanas. She even has a Welsh (or sometimes Scottish)-sounding accent, which is likely the result of being raised by a Scottish priest.
The Vatican's Iscariot Section XIII consists almost exclusively of orphans who were raised by Father Anderson, so it's not surprising that members would come from all over the world. And it's shown that Anderson (who's a bayonet-wielding Knife Nut) gifted all of his students with their individual weapons, so Yumie's katanas are probably a homage to her Japanese heritage.
A variant, unrelated to use of special powers: In Mai-HiME, there is a small Roman Catholic church seemingly staffed only by a single priest named Father Joseph Greer (probably intended to be American) and a single Japanese nun, Sister Yukariko Sanada, with no other "functionaries" of any kind. This setup is a lot more plausible in a Shinto temple than in a church. In the end, Sister Yukariko is seen no longer in a habit, very pregnant, but somehow still in charge of the two new nuns, Miyu and Nao, who are part timers while they still go to school (Also, one of them is a Robot Girl. You have to wonder about the theological implications...).
Similarly, the Gundam Wing manga Episode Zero shows that Duo Maxwell, who inexplicably wears a priest's collar in the series, once lived at a church staffed by a single priest (Father Maxwell) and nun (Sister Helen), who are killed in the war.
Misora, Cocone, and Shakti in Mahou Sensei Negima!, who wear ridiculously short habits. Then again, they do have to fight in those things, so a shorter length may be preferred. In one chapter, she was put in an even shorter habit, and was less than pleased about it (if only slightly so). That said, the church they are part of at least superficially resembles an actual church; there's a Confessional, at least. It doesn't get used like a confessional is normally used, but that's more because the girls are unfamiliar with it, and because a mischievous nun (Misora) was on the other side.
Trinity Blood features a number of women who are apparently meant as nuns of a sort, including the very well-endowed, overtly seductive Noélle Bor and the demure — yet-gun-toting — Esther Blanchett. Oh, and on a related note, the series also features Catherina Sforza, a female cardinal.
In Witch Hunter Robin, there's a male example — Robin's mentor, a Catholic priest, has a daughter. While this is possible note Widowers are allowed to become priests, as are married Protestant ministers who convert to Catholicism. Additionally the Eastern Catholics, who make up 2% of all Catholics, allow married men to become priests. However Eastern Catholic churches have very distinct iconography that would be obvious. it is very rare.
In School Rumble, Yakumo's British friend Sarah Adiemus appears to be a nun on an after-school, part-time, volunteer basis. Part of her duties include hearing confessions, which is actually only done by a priest or bishop. And we later find out that other teens periodically come into the confessional to ask for relationship advice.
Averted in El Cazador de la Bruja, in which the two protagonists shelter in a South American convent with genuine nuns. Admittedly, it was an unusually small convent with only three nuns, and one can only hope that their order's vows didn't include non-violence, but the presence of an aged Mother Superior ensured that they couldn't really be analogous to mikos.
Yumina from SoraNoWoto is a Miko-Nun living in a Shinto Church, I mean just look◊. It is interesting to note, though, that Yumina acts more like an actual young Christian nun than most anime "nuns", providing spiritual support (or trying to) and running an orphanage.
The fact that the series's world is a hybrid of multiple cultures don't help.
In an early chapter of D.Gray-Man the brother of an episodic character is a priest who was going to be married to a nun only to have her killed pages in.
The title character of To Aru Majutsu no Index is a very young un-nun-like nun who others politely refer to as "Sister-san." She serves the Church of England rather than the Catholic church. Regular nuns in universe are an organization of female spell casters.
Michiko to Hatchin has the male equivalent in Hatchin's foster-family from Hell. Her foster-father is a Catholic priest living with his wife and two children.
In Kannagi, Zange acts as one since her host body, Hakua, is the daughter of a Catholic priest. No, she's not some illicit lovechild. Her being a daughter of a priest is treated in the same way as daughters of Shinto priests, who are not required to be celibate.
10-year-old Maria Takayama (who's also a teacher) and her older sister Kate in Boku wa Tomodachi ga Sukunai are both nuns, but the way their positions work seem much closer to being mikos, since their family's women don the habit as a long-standing tradition and it's treated like more of an after-school job. Later it's revealed that Maria is neither officially a nun or a teacher; Kate just brings her to school with her because there's no one to look after her at home, and she makes Maria wear the habit because it's cute.
In an anime-inspired western example, the comic book series "Warrior Nun Areala" by Ben Dunn.
Most of the cast of La Pucelle Tactics. It is justified in that one of the cast is the founder of the church, and as such could make whatever rules for the order he wanted. It should also be noted that this is not standard in the game world: La Pucelle is a special order of demon hunters belonging to a specific church; it is the only church with such an order and not all of it's members are initiated into La Pucelle.
Harvest Moon plays with this. Their religion isn't Christianity, but it's a mix of Buddhism and Catholicism. You can't marry any of the priests or nuns, but one of the games teases you by letting you court a nun... Only for her to turn you down.
This is averted in Harvest Moon: Twin Villages (known as Tale of Two Towns to the Western audience), where you can actually marry the game's nun-like character, but only after gaining permission from the Goddess she serves.
Ciel from Tsukihime. However, the organization she belongs to (the Burial Agency) is arguably the most heretical part of the Church, where faith is secondary to actual power. She does dress up as a nun on occasion, but she says that it's all for show and doesn't actually mean anything.