There are many questions: How to cross the TRNC border correctly? Do they put a stamp on the passport or not? What are the after-effects of the NC stamp in it?
So, at the border control, you have two options - a stamp on your passport or a form. If you do not want your arrival in the northern part to be official, you need to take this form next to the border control and hand it in along with your passport - they will stamp it on this form.
The official stamp of NC in your passport can only harm you if you want to fly to the southern part of the island through the airport in Larnaca or Paphos in the future. For other countries, this stamp does not matter. But for the Republic of Cyprus, an official crossing the border of Northern Cyprus is a violation of internal laws, and they may not put a national visa of the Republic of Cyprus in such a passport. Therefore, to travel freely, experienced travelers are advised to have a second passport. One is for Northern Cyprus, and the other - is for the southern part. As practice shows, you can get a residence permit from NC and the Republic of Cyprus simultaneously with two passports.
Also, remember that if you arrive in Cyprus through the airports of Larnaca or Paphos, you should leave from your airport of arrival. Unfortunately, entry through the southern part and exit through the northern one may cause further troubles in the Republic of Cyprus upon re-entry. According to the country's legislation, if you leave through Northern Cyprus, the Republic of Cyprus does not fix it in any way, and you appear to be an illegally remaining foreigner who overstayed his visa. But if you have a passport from one of the EU countries, you can travel without obstacles and use any airport on both parts of the island.