In the Opening scene where Racine is walking with his business partners talking about "selling ice to Eskimos", Racine enters the taxi-cab with his trench coat on. When he arrives at the hotel, he gets out of the cab without the trench coat. He leaves the hotel on a stretcher, still without the coat, but it appears again when he puts it on as he tries to leave the hospital.
At the start of the final ninja/samurai battle, Kinjo is seen drawing a sword from a scabbard on his back. Over the course of the fight, the scabbard disappears, reappears with another sword in it, then disappears again.
Paul Racine escapes his ninja pursuers by ducking into a pachinko parlor (a kind of Japanese gambling hall). He is aided by a young girl, who takes him to Nagoya Station, where he catches a 5am bullet train out of the city. It's illegal for anyone under 18 to go into pachinko parlors, so the girl wouldn't be there in the first place. Pachinko parlors close at 11pm, meaning that Racine would be in for a very, very long wait on arriving at the station.
In one of the shots where Takeda and Racine spar (after Takeda receives his new sword), a camera man is clearly scene on the left edge of the shot. (at around 26 mins)
While Takeda is speaking with Oshima about making a new sword, a Stellar's Jay can be heard in the background. Stellar's Jays are a bird found in North America, and are not found in Japan.