The reason Asuka wears an eyepatch is due to being contaminated by the Ninth Angel in the third film. It is a reference to a memorable scene from the original series where Asuka is pierced by the Lance of Longinus. The eyepatch also symbolizes the passage of time since the events of Evangelion 2.0. Here is an image of Asuka wearing the eyepatch.
Pirate Asuka made her first appearance in the previews after the second film, and was finally featured in the third film. Fans were left wondering how she ended up losing an eye. The second Rebuild of Evangelion movie teased the next film, which wouldn't be released for another five years. It showed Asuka wearing an eyepatch, leading to speculation and jokes among fans. The fate of her eye is briefly explained in the third film, in which she pilots Unit 03. The EVA gets contaminated by an Angel, causing an emergency, and Asuka ends up losing her eye during the incident. In the next film, she is seen wearing an eyepatch, though it is not specifically addressed. Later in the film, she attempts to stop Shinji and Kaworu with her upgraded Unit 02 in Terminal Dogma, but is unsuccessful.After exhausting all other options, Asuka removes her eyepatch, unveiling a "sealing pillar" implant where her eye used to be. Surprisingly, the Ninth Angel was never completely eradicated from her body. As she removes the seal, the Ninth Angel regains control, causing Asuka (along with Unit 02) to transform. Unfortunately, this outcome was part of the plan, leading to the absorption of Asuka's Angel core (the entry plug) into Unit 13.
Neon Genesis Evangelion's EVAs are one of the most distinctive elements of the series. Here is a breakdown of each EVA encountered in the original anime.
From a broader perspective, the reason behind Asuka wearing an eyepatch has another significance. In End of Evangelion, Asuka and Unit 02 are defeated by Mass Production Evas armed with replicas of the Lance of Longinus. One of them hurls the Lance at high velocity, piercing Unit 02. The eyepatch in Rebuild of Evangelion serves as a homage to this iconic scene and perhaps acts as a trick to make diehard fans believe they know the fate of her eye, only to be surprised by the revelation of the Ninth Angel. Additionally, it visually implies that a considerable amount of time has passed, despite the lack of aging among Asuka and the other Eva pilots.
The eyepatch reveal indeed spawned humorous internet memes, but Neon Genesis Evangelion has skillfully transformed what initially seemed trivial into something truly horrifying, much like the series excels at doing.
You can catch all Rebuild of Evangelion films on Prime Video. Enjoy streaming!