Half-way between identical twins and fraternal twins, there is one other possibility.
Polar body twinning occurs when a female's egg splits in twain prior being fertilized, and each half is fertilized by a different sperm*.
While they are classified as fraternal twins, they share 75% of their DNA (They share 100% of the DNA from their mother, and at least 50% of the DNA from their father), while most fraternal twins share only 50% their DNA, the same as regular siblings. Because of this they look more alike than do most fraternal twins, and are sometimes called 'half-identical twins'.
* It is assumed that the two sperm are from the same man. If this is not the case, then you would presumably still refer to it as a case of polar body twinning, but not half-identical twins. The twins might then share only 50% of their genetic material.