Now I was already vaguely familiar with the tune, but had never really parsed out its meaning. To my great surprise, it turned out to be the song sung in a famous scene of The Bicycle Thief, where father and son enjoy a momentary respite from their hellish situation in post-War Italy.
I had always been vaguely intrigued by the unidentified song, and wondered why adorable Bruno laughed at it. It turns out that the politically incorrect song is about the number of Neapolitan women during the occupation who gave birth to black (or actually biracial children). The amusing refrain states that the mom, despite the baby's different appearance, goes ahead and calls him Ciro, or Antonio, or Peppe, or Ciccio. Typical Neapolitan names. Everyone pretends that they can't imagine what happened, despite the large number of black or brown soldiers openly fraternizing with the local ladies.
It's not so easy to find translations into Italian, much less English, of the song. Perhaps the Neapolitans have kept it to themselves, as it indirectly brings up the painful chapter of the widespread prostitution between soldiers and inhabitants at that time. But an Italian translation can be found here.
And here is an excellent rendition of the song in its entirety. Note that tammurriata refers to a type of song accompanied by a tambourine. So this would be the black tambourine song.