The Vikings, the group often referred to as Normans, established a presence in Sicily and southern Italy in the 11th century. Being social people, I presume they socialized with the local population. As did Arabs who arrived later.
I think that the Middle Eastern features that you see in many Italians may go back even further: to the time that Phoenicians and Carthaginians colonized the area.
The Arabs arrived first; after some considerable period of time, mercenaries were hired by the Byzantines to drive them out. After that, some Normans arrived and carved out a short-lived kingdom in Sicily and if memory serves a small chunk of the Italian boot. Wanting the best for his kids, King Roger (”fweee Woger!”) hired Arab tutors and the whole thing went all kumbaya after that.
The (purported?) tomb of Hrolf the Ganger has a very-postmortem inscription referencing the various dynasties which had sprung from him, and the Sicily kingdom was one of them.