North Cyprus operates a fairly modern state run healthcare system, albeit still a little behind the times in places. However, it is perfectly acceptable for the majority of the population, although expatriates may also prefer to have the addition of private health insurance if they can afford it.
First things first, and that is North Cyprus is at present not a member of the EU, and therefore the European Health Insurance Card (EHIC) will not be valid at any medical facility here, whether state or private.
