At the time that these Genoese (called Ligurian by one source) and Venetian sailors would have shipwrecked on the western coast of Cuba (1605), the province was referred to as Nuevas Filipinas or "New Philippines" due to the transfer and connection with the tobacco plantations in the Philippines and the huge influx of Filipinos, both caused by the Manila Galleons (they sailed every year or twice a year between Manila to Acapulco from 1565-1815). It was in 1774 that the province was renamed Pinar del Río which translates literally into "pinewood of the river", referring to the pine forest found along the River Guamá.
However, in regards to Mantua, its founding by shipwrecked Italian sailors (who landed at the coastal location known as los Arroyos de Mantua) who wandered inland and created a settlement is an oral tradition passed down through the generations by the townspeople. I recently looked up sources about the subject and one confirmed that, around that time, the Pope had stated in a document regarding an Italian ship, either named Mantua or had some relation with Mantua in Lombardy, being confused for pirates by the English and being pursued until it crashed into the coral reefs. Another source suggests the possibility that the ship was ordered into a secret expedition to the New World by Duke Vincenzo I Gonzago of Mantua. Either way, the settlement became an official town in 1716 under the name of Guane del Norte and later came to be officially called Mantua. Currently, Italian surnames, such as Ferrari, Pitaluga, and Fiorenzana, are found there. Also, a source noted the lack of Italian surnames recorded in Mantua during the Cuban revolution by General Maceo, but it was because he was received by the Spaniard population of the town, who were more well-off, while those of Italian descent (the fishermen and farmers) were not documented.