Before we officially "meet" Tyler, when our main character sits next to him on a plane, we see him in a series of subliminal flashes, like near a copier or behind a doctor. This is our first clue that Tyler is not what he seems.
There are tons of clues to Tyler's true identity as a figment of our narrator's imagination throughout the film. One of our main character's first remarks to Tyler is "We have the exact same briefcase." Tyler smirks because he knows that it is the same briefcase…because they are the exact same person.
Tyler Durden is an anarchist imaginary friend whose purpose, ironically, is for our main character to be more honest and real with himself. Our narrator uses sarcasm as a defense mechanism, and Tyler immediately shoots him down: "How's that working out for you? […] Being clever?"
In order to keep it real, Tyler blows up our narrator's apartment (which means he blew up his own apartment), forcing him to move in together (i.e. move in with…himself) and work odd jobs. One of Tyler's jobs is projectionist:
TYLER: Someone has to be there to switch the projectors at the exact moment that one reel ends and the next reel begins […]
This job which serves a dual purpose: 1) It enable Tyler to splice porn into family films. 2) It's commentary on the fact that Tyler is the movie's projectionist, sending us subliminal hints and tricking us, through the storytelling, into not realizing that he and the narrator are the same person until the end. Clever girl.
As things get more extreme—Project Mayhem blows up buildings, policemen are threatened with castration—our narrator questions Tyler's motives. Or his own motives. Whatever. Tyler tries to convince him otherwise, saying, "Stop trying to control everything, and just let go. Let go!" At this point, our narrator literally lets go of a car steering wheel leading to an accident (that car really needs it alignment checked) and figuratively lets go, Frozen-style.
Unfortunately, when Jack/narrator finds out that Tyler is planning on blowing up buildings, that's going too far. As he starts to realize just how deep Tyler's plan goes, he seems to be more concerned with eliminating this uncertainty within him, this entire personality he created, no matter what the plan may be.
TYLER: You were looking for a way to change your life. You could not do this on your own. All the ways you wish you could be, that's me.
But our narrator decides, for unexplained reasons, that this isn't who he wants to be, despite having no problems going along with the plans for the entire movie.
When our narrator realizes he is Tyler, Tyler starts appearing crazier than ever. Brad Pitt shows up with a shaved head, a chipped tooth, and a fur coat out of a Macklemore video over a mesh shirt. Oof. Poor choice, Tyler.
Here, Tyler briefly transforms into the movie's antagonist, even though the antagonist had been society for the previous two hours. Instead of Man vs. Society, we now have Man vs. Himself in both its purest and most wacked-out form.
Tyler is "killed" when our narrator puts a gun in his own mouth and pulls the trigger. But do you think Tyler is really dead, or does he still live somewhere deep inside Jack?