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Definition of "Half-identical Twins" from Wikipedia:
Half-identical or semi-identical twins (also referred to as "half twins") are the result of a very rare form of twinning in which the twins inherit exactly the same genes from their mother but different genes from their father.
Although examples of half-identical twins have been found, the exact mechanism of their conception is not well-understood, but could theoretically occur in polar body twinning where sperm cells fertilize both the ovum and the second polar body.
This situation is not the same as the common form of fraternal twinning, in which two genetically different ova are fertilized by two genetically different sperm. In this case, the ova are genetically identical.
There are two mechanisms by which this might happen:Polar twins (or "polar body twins"), where two sperm fertilize an ovum, one of the two fertilizing a polar body; or where an ovum splits into identical copies, one containing a polar body, prior to fertilization, allowing it to be fertilized by two different sperm.A 1981 study of a deceased triploid XXX twin fetus without a heart showed that although its fetal development suggested that it was an identical twin, as it shared a placenta with its healthy twin, tests revealed that it was likely a polar body twin.
Sesquizygotic twins, where two sperm fertilize the one ovum, forming a triploid, and then splitting.
The authors were unable to predict whether a healthy fetus could result from a polar body twinning. In 2003 a study argued that many cases of triploidity arise from semi-identical twinning.
In 2007, a study reported a case of a pair of living twins, one a hermaphrodite and one a phenotypical male. The twins were both found to be chimeras and to share all of their maternal DNA but only half of their father's DNA. The exact mechanism of fertilization could not be determined but the study stated that it was unlikely to be a case of polar body twinning.