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troperville

So, you just happen to be the child of that army guy who went native and married The Chief's Daughter? Or perhaps the reverse applies and dear mum left home and joined dad on the ancestral family homestead back in rural Smalltownington. This of course is assuming both your parents are alive and love each other, (Star-Crossed Lovers have a tendency to have it rough), or you might be the product of less than consensual affairs. And then you were born.

Please note that in Real Life, the term "half-breed" is a highly offensive slur, which is why this trope got renamed.

These tropes all relate to the concept of Mixed Ancestry:

Examples:

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Anime and Manga

In Sora No Woto, it is implied that Rio is of mixed heritage since she lacks the Phenotype Stereotype looks of her blonde, green-eyed half-sister and her last names are Japanese/Slavic. Most probably a very common thing in Helvetia and no one seems to care.
Urd from Ah! My Goddess is half-demon on her mother's side. Off course, the only one the really seems to be affected by it is her as most goddesses are nice enough not to bring it up (or simply don't care) and most demons, well they don't really want to mess with the beloved daughter of the ruler of Hell herself.
Blue from Wolf's Rain is a wolf-dog. Upon meeting Cheza, the dog side of her virtually disappears and she joins the other wolves.
Area 88 manga:
- Saki was the son of an Asranian prince and his Greek wife. Fortunately, Saki suffered no stigma in Asran on account of his mixed heritage.
- Not so for Josie, an orphan of mixed French and Japanese ethnicity. After her parents' deaths, her Japanese grandparents refused to take her in because they disapproved of her parents' marriage.
Maka from Soul Eater is a the daughter of a Technician and a Weapon. Being a Weapon is a genetic trait that Arachne created by mucking around with witch and human souls 800 years ago, so they aren't really a separate species.

Comic Books

The DCU:
- Supervillain Cheshire (Vietnamese/French), who basically went insane.
- Mongrel, a.k.a. Josh Xan, was half-Vietnamese as well. He died in Infinite Crisis.
In Gold Digger, the platinum dragon D'bra is actually a hybrid between two differing tribes of dragons, platinum on her mother's side and iron on her father's. This is notable as full dragons are rare enough between two partners of the same tribe (most eggs hatch into less drakes or wyrms), and hybrids are thus extremely powerful. However, not only are they extremely rare (six have been recorded in the entirety of draconic history), but also card carrying demonstrations that With Great Power Comes Great Insanity. D'bra's own explosive temper is a considerable worry to the other dragons who are afraid of her snapping like all the others.

Film

Elvis Presley plays a man with a white father and an Indian mother in Flaming Star.
Frank Hopkins in Hidalgo is half Native American and half white, and has a foot in both cultures.
In Indiana Jones and the Last Crusade, one of the treasure hunters in the prologue is credited as "Half Breed".
The main character, Zen, in Chocolate is the child of a marriage between a Japanese Yakuza and an enforcer in a Thai mob.

Literature

Rehv, from the Black Dagger Brotherhood series, is half-vampire and half-sympath.
Oliza in Amelia Atwater-Rhodes' Kiesha Ra series is a half-snake shapeshifter, half-hawk shapeshifter. She has the forms of both a hawk, a cobra, and a combined form referred to as a wyvern. Her cousin, Hai, is half-falcon and half-cobra and suffers from unstable magic that allows her to see several possible timelines (making her a Waif Prophet) but leaving her starkly out of touch with reality. Oliza and Hai are also both example of All Genes Are Codominant
In Graham McNeill's Warhammer 40000 Ultramarines novel Black Sun Dead Sky, Honsou is termed a half-breed. In Storm of Iron, it is revealed that this is because he has gene-seed stolen from Imperial sources as well as the Chaos ones; the Warmaster exploits this to develop rivalries.
- Honsou's gene-seed isn't just from an Imperial source - it's from the Imperial Fists, the Iron Warriors' arch-enemies. As a result, calling him "half-breed" remains his Berserk Button for quite some time.
A number of characters in J. R. R. Tolkien's The Silmarillion and The Lord of the Rings are mixtures: L�thien (half Elf, half Maia), her son Dior (half Human, 1/4 Elf, 1/4 Maia), E�rendil (half Elf, half Human) and their various Heinz Hybrid descendants (Elwing, Elros, Elrond, Aragorn, etc.). All seem to be well loved and respected by both sides of their respective families.
H. P. Lovecraft has the Whateley Twins, Wilbur and OH MY FUCKING GOD WHAT THE HELL IS THAT?! Tends to happen when the mother is human and the dad is an Outer God.
The Ysabel Kid, from The Floating Outfit novels of J.T. Edson, is half Kentucky Irish, a quarter Commanche, and a quarter French-Canadian.
The Ewu in Who Fears Death, being born from the union of Okeke and Nuru people. Looking surprisingly like neither tribe, the Ewu are sand-colored, skin and hair, with light eyes, like albino Africans.
Some totally non-human examples from Animorphs:
- Jara Hamee and his daughter Toby (named for Tobias) are descended from an Andalite female who married a Hork-Bajir, though, again, she used morphing technology to actually become a Hork-Bajir.
- The Ellimist had children while living among the Andalites in a genetically engineered body millions of years ago, so a great deal, if not the entirety, of modern Andalites should be his descendants. Which means Tobias and the Hamee family are his descendants, too. And indeed, it is implied the modern Andalites inherit their built-in Translator Microbes (i.e. "thought-speak") from his artificial body. The primitive Andalites he lived amongst had only sign language.
Terry Brooks' Magic Kingdom Of Landover series has Willow, born of two different kinds of faerie, whose status complicates both her relationship to Ben Holiday, and later their child's gestation and birth — as the child of a wood nymph, she has to spend part of the time as a tree.
There are a few examples of hybrids in Terry Brooks' Shannara books (the Urda, for example, are Troll/Gnome hybrids). Justified, in one sense, by the fact that dwarves, gnomes, and trolls are simply different subspecies of human mutated by the apocalypse. Less explicable is how humans can interbreed with elves, who are just fae evolved into humanoids...
In Smoke Over Grimsvotn, the Demoness of Smoke, Lilitu Geisthexe is strongly implied to be the daughter of the Goddess Urd Jotundottir. Urd herself refers to the Queen of the Demons as Mother, making Lilitu 3/4ths Demon biologically.
- More central to the plot of the actual novel, the children of Mao Semezou are Demi-Gods.

Live-Action TV

In Charmed Paige is a half-witch, half-whitelighter. She gains whitelighter powers such as orbing and healing without having to actually die though she has to learn how to use them while whitelighters know them automatically. Wyatt and Chris are also the same though stronger because their mother is a Charmed One.
Abed Nadir in Community is the son of a Palestinian father and a Polish mother.

Newspaper Comics

The titular hero of Latigo, by Stan Lynde, was the son of an Irish father and Native American mother.

Theatre

Azuri in The Desert Song.
One of the love plots in the musical South Pacific (also a film).

Video Games

This trope is played up in the video game Red Dead Revolver with Red, the main character, being the offspring of a white settler and a Native American.
Connor, the main character in Assassins Creed III, has a similar background.
The Final Fantasy series falls into this:
- One of Reveals in ''Final Fantasy IV is that Cecil Harvey and his brother Golbez are half-Lunarian. Cecil's son Ceodore is a quarter-Lunarian, owing to Cecil's marriage to the human Rosa.
- In Final Fantasy V Bartz is fully human, but his parents hail from different halves of the world that existed as separate planets for centuries.
- In Final Fantasy VI, Terra Branford is half-human and half an esper, a race of powerful magic beings who are not always humanoid. Most of the first half and her character arc revolve around her powers and identity struggle.
- In Final Fantasy VII, Aerith had a human father and a Cetra mother. This is also highly significant to the plot.
- In Final Fantasy X, Yuna is the daughter of a normal human and an Al-Bhed, a human ethnicity distinguished by their green eyes and spiral irises. The mixture manifested itself as Yuna's Mismatched Eyes.
In The Elder Scrolls, a book called "Notes on Racial Phylogeny" states that children of interracial unions typically resemble the mother more, with some traits from the father still persisting. In Oblivion, the Gray Prince, a champion gladiator is part human, part orc^note. The entire race of Bretons descend from hybrids between nedes (ancestral race of men) and Aldmer (ancestral race of elves), when the Elves took concubines. The Bretons are considered a race of men, but their elven ancestry manifests as having the highest racial aptitude for magic of all men races. A few aristocratic Breton lineages still retain a slight point to their ears.

Web Comics

Banquo of Goodbye Chains. His character profile proudly describes him as "half-Mexican and half-Swedish, and also a colossal dick." He was the result of a one-night stand between an actress and a mestizo Mexican man, and faces quite a bit of racism because of his background. His Distaff Counterpart and part time lover, Cordelia, is half-Ute, but there are certainly no parallels between them and how they treat their companions, no siree.
Antimony Carver from Gunnerkrigg Court is a (mostly human) human/fire elemental hybrid, which is lampshaded by Coyote.
Coyote: What an interesting first union that must have been...

Western Animation

Similar to the DCU example in Comics, in Young Justice, both Cheshire and her sister Artemis are half-Vietnamese, half-white. Interestingly, Cheshire comes across as much more Asian than Artemis in terms of appearance, style, weapons, and voice. Semi-justified by the fact that Cheshire has explicitly rejected her white father; she goes by "Jade Nguyen," not "Jade Crock," while her daughter is "Lian Nguyen-Harper."
Increasingly common in The Legend Of Korra. The original series had very few mixed-race couples and no mixed race children, but in Korra, due to the nature of Republic City many characters are of mixed race. This is most interesting since Bending is partially genetic, and one person can only have access to one type of bending, leading to brothers with completely different abilities.
According to Word Of God, Ambiguously Brown Mike Chilton from Motorcity is of mixed race.