Revealing mistake: Right after Father Alvito hands Blackthorne the Portuguese-Japanese dictionary, the Jesuit priests leave Blackthorne and his entourage to continue their journey onto the port of Nimazu. You can see the orange-robed clerics walk along a track that was obviously used by wheeled vehicles. You can see grass growing in the middle of the dirt road and none in the grooves. But in an earlier scene Father Alvito stated that only the Emperor was allowed to drive in a wheeled cart which he only did it on rare occasions for ceremonial purposes and for short distances as this would otherwise damage the roads (as the grooves testify).
Factual error: When they first meet, Rodrigues tells Blackthorne that aboard his ship there is plenty of food and grog. The problem is that grog was not "invented" until 1740, when it was introduced into the British Navy by Admiral Edward Vernon, who was nicknamed "Old Grog" because of his habit of wearing a grogram coat.
Other mistake: After Lord Buntaro's archery demonstration, Blackthorne comes over to the post which now bears the three arrows shot by the Japanese. In the paper wall behind him a hole can be seen, which is to be the spot through which the arrows have come. but the arrows and the hole do not make one straight line, which would have to be the case if the missiles were to really have arrived from there.