Death Note: The Last Name

[Lawliet]

At AB, when watching Death Note: The Last Name did this occur to anyone else? If the Death Note Misa had was fake, since Watari had swapped it earlier with the real Death Note the second day of Misa's release, how can Light (Kira) see Ryuk after he touches it when Misa gives it to him near the end of the movie. btw this occured to me while I was watching it again a couple nights ago, but ya.