What the adherents of the manosphere and Patriarchy fail to understand is that it, not the world under the influence of Feminism and "wokeism", is The Matrix, the thing that inspires the feeling that something is wrong with the world. Their doctrines and ideals are caricatures built on falsehoods, miscommunication, misconceptions and misunderstandings. They're not taking the red pill; the feminists and LGBTQ+ communities and allies are. (The red pill contains Oestrogen hormone replacement supplements. It looks like how they looked in the 1990s on purpose.) If anything, they're taking the blue pill, a placebo (or maybe it's the viagra they think they need), and trying to stay in their warped and ultimately detrimental fantasy land. They're too busy paying attention to the woman in the red dress to have heard anything Morpheus said.

Neo: Jesus! I thought you were a guy!
Trinity: Most guys do.

Keanu Reaves has confirmed that, in early/original versions of the script, Switch is trans* (presents as a man in The Matrix, but as a woman in Zion, IIRC), hence the name Switch. It's a core part of the character arc, but it was changed because the producers/studios found it too difficult/radical a concept to sell. Chalk it up to yet another glitch in The Matrix.

If you're still in doubt, I suggest you watch the films again, with a critical eye. It's full of symbolism (both subtle and blatant) that uses typical film techniques to subvert the classic format and expectations of male-dominated cinema.

• Neo lives a double life, with a feeling that something is "off" about the world, a persistent nagging feeling he can't explain. According to what I've read, this is common to trans* people. Not that I have any doubts about my gender identity or am going to presume I can relate, but I have that feeling about masculinity.
• It's the women, not the men, in the Nebuchadnezzar crew who take Neo to his first meeting with Morpheus.
• After a scene in which Cypher looks Trinity up and down, there's a cut to a red "don't walk/stop" traffic light in line with Cypher's gaze.
• After ogling the woman in the red dress (a decidedly male perception/portrayal of femininity and the male gaze), Morpheus advises Neo to "look again". There's a gun (a decidedly masculine/phallic object) pointed directly at Neo.

I'm sure there are other examples, but those are the ones that come to mind.

"Neo has dysphoria. The Matrix is the [false] gender binary. The agents are transphobia. You get it."
 — Andrea Long Chu; "What The Matrix Can Teach Us About Gender" (2019/02/07); Vulture

Fight Club

Marla Singer [commenting on the dildo next to her bed]: "Don't worry about that. It's not a threat to you".

Tyler Durden, with his toxic masculinity and posse of disaffected blue collar workers, is not the hero of Fight Club; Marla is. Tyler dies before the end of the film, remember? For all that it appears to glorify machismo, violence and taking down the established world order through it, Fight Club is really about embracing the softer, more "feminine" aspects of one's nature. I'm not sure I'd go so far as to call it Feminist, though (but that's because I don't know enough to take that stance). The Matrix, on the other hand, is definitely Feminist/trans* in its themes, a critique of Patriarchy and transphobia.

Jack/Narrator [Voice-over]: "Strangers with this kind of honesty make me go a big rubbery one."

Well, that's an odd thing for a supposedly typical manly man to say, isn't it?

I think it's time the manosphere reassesses its takes on pop culture and rethinks its philosophies. It seems to me it's time to go back to the chair and plug in again (but instead of drunken kickboxing, it learns some social skills and a few lessons on "women's issues"). The Tate brothers are not the droids you're looking for [sic]. Just a thought ...