While easily missed, the way to get this conversation is quite simple:

1. Choose to work at the library.

2. Continue to play the game as though you are pursuing Nageki's ending until the game gives you the option to "wish for the mad love of a fallen angel" at Tanabata to begin Anghel's route. (If this option does not appear, you haven't unlocked Anghel's route yet.)

3. Follow Anghel's path until your character decides to "be a Cultural Girl" and visit the library. When you talk to Nageki, Anghel will interrupt. Enjoy the resulting conversation!

While this section deals with the video clip above, it also deals with major spoilers for the BBL route as well as Nageki's ending. If you haven't played the entire game, including the "true route," do not read this section.

Once you've played Nageki's route, and especially after playing the BBL route, this optional conversation has several interesting points.

Anghel can see Nageki. No other character in the game interacts with Nageki other than the player until BBL. Why Anghel can see him is unclear, but it's an interesting point.

This conversation contains foreshadowing for the BBL storyline, first with the reference to the 12 missing encyclopedias that will come up later as a part of the mystery.

More importantly, Anghel addresses Nageki as "Textoris Melodia Funus." This is almost identical to his name for BBL-time Ryouta, "Textoris Melodia Funeris."

These two names translate to "weaver of requiems" or "weaver of funeral melody." That Anghel addresses both characters this way suggests he can either sense the Charon Virus or knows more about what's going on than anyone else realizes. Keep in mind that his archnemesis in his character route is Shuu.

Either way, it's a cool bit of foreshadowing for the BBL revelations.