That is, to get to the Steak scene?
The excuse answer : The _______ Allows it. [Oracle, Merovingian, Agent Smith]
The answer I think we are going to get : Smith Allows it. How Cypher logistically gets the cables into his body, that’s something else.
Alternate : If Mouse [IIRC] can spend the day dating The Lady In Red, I would assume Cypher gets just as much “free” time in the matrix. How Tank and Dozer miss this, that’s something else.
Random thoughts :
To the OP: I’m going to go a few steps beyond FinnAgain : If bodies popped out as fast as people die, the pill scene would have left something to be desired. [I think someone up-thread got the colors mixed up.]
Neo obviously has a tough choice on his hands as he meets Morpheous. If he could simply recall a body popping out of existence the choice would be fall of a log simple. ““Sure, I’ll Go””. But, I think the viewer is forced to accept that bodies stay in. The entire idea of the desert of the real would fall flat otherwise. The “Epicness” of the choice wouldn’t be there.
Also, Morpheous explains that we are living in a dream world, as we knew it at the peak of our civilization. I believe he uses “End of the 20th Century” or some such. Using 1999 as our guide [for the given reason, and the obvious reason]** I think we can conclude that as 1999 had cold bodies**, morgues, cadavers, and lifeless bodies found in a bathtub, bedroom, in the woods, etc, **so too would The Matrix. **Bodies stay put. Until the funeral. And yes, there would be a full funeral. Again, nothing out of the ordinary, nothing to suggest it wasn’t 1999.
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After Neo Takes the pill, Cypher looks into a goggle apparatus [He delivers the Kansas is going bye bye line] I would assume that Neo ““dies””, as a normal person would see it. Neo takes the pill. He essentially shorts out his pod. He gets ejected. The Neb then comes around and plays the claw game with him.
After you log out, I think the Avatar drops out. We wouldn’t have the need for the Residual Self Image speech in the construct.