Every cell in the human body has proteins on the surface, some of which are used by the immune system to identify “friend or foe” for the cell. The human blood type is made of a set of the ones used by the immune system. As such, an incompatible blood type will result in an immune response, where the immune system decides the blood is an invader and begins attacking it.
The actual result will depend on the amount of incompatible blood transfused. With very small qualities, nothing too bad will happen, but the transfusion will be useless. The breaking down of the blood cells, however, increases the risk of blood clotting and releases various toxins into your blood. This can result in severe blood clotting, stroke, uncontrolled bleeding, and renal failure. These problems can be treated if caught quickly enough, though they are very dangerous and blood transfusions are typically given to already weakened patients.
It is worth noting that the common ABO/Rh blood system (where we get types like O+ and AB-) most people know is incomplete. Human blood has [51 different](https://en.wikipedia.org/wiki/Human_blood_group_systems) *known* blood type components, though many are fairly irrelevant*. Due to this, doctors will perform a final test with a small sample of the patient’s and donors blood, mixing them together (called [crossmatching](https://en.wikipedia.org/wiki/Cross-matching)) and checking for a reaction. Only if no reaction can be detected after a short time, will they actually give the transfusion. In emergencies, the transfusion may be started before this test is complete, though typically blood already tested for maximum compatibility will be used.
* Some of these other ones just don’t matter medically. Others are merely super rare. Also, note that I included A and B as two separate components, though they are normally treated as a single system – O is merely the lack of both A and B.
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