In The Neverending Story, Michael Ende faces a somewhat similar challenge: the main character, Bastian, gets his hands on a book, and the narrative alternates between the book Bastian is reading, and his own actions - his thoughts with regards to the book, his more mundane actions with regards to skipping school.
Michael Ende solves this beautifully by using two colours: one narrative line is printed in green, the other in red. (And the book Bastian is reading is also printed in red and green, making the whole thing meta.)
If you do not wish to use colours, I would turn to using significantly different fonts. (Which is incidentally how paperback editions of The Neverending Story are printed.) While fonts are more subtle, they still provide a strong visual cue to the reader regarding which layer of the narrative they are currently in.