The Eye in the Pyramid came out in fall 1975 as a Dell original paperback. While at this point it is impossible to know in detail which parts of Illuminatus! were written by Robert Shea and which by Robert Anton Wilson, a fair amount is actually known about how the work was written. They took turns writing sections, with Shea producing much of the "melodrama" and Wilson much of the satire. Robert Shea's parts were rewritten by Wilson so that the work had a uniform tone. Shea wrote the Atlantis section. Wilson supplied the Lovecraft/Cthulhu Mythos material, the occult material, the references to James Joyce, the signs from the Midget, and, very likely, the Burroughs cut up technique.
Wilson and Shea were both very familiar with anarchism/libertarianism, both knew Discordianism well, both knew Chicago and New York well, both were ex-Catholics familiar with Catholic priests.
The editors at Dell trying to get the work published found Shea easier to work with than than Wilson; for example, Wilson resisted dividing the work into three volumes. See this interview with Dell editor David Harris: "Wilson was not much interested in cooperating with any of my ideas until I put it to him that I was his only friend in the company, and that if I dumped him his manuscript would be put on a shelf to die. (Actually, I think I said that it would be tossed in the East River.) He finally agreed to the three-volume idea, and I got it into the schedule not long before I left Dell, which would have beHen October of 1974."
Here is more detail and some citations from interviews on how Illuminatus! was written.
Here is Wilson on who wrote what: "In general, the melodrama is Shea and the satire is me; but some of the satire is definitely him and some of the melodrama is certainly me. 'When Atlantis Ruled the Earth' is 99% Shea. The sections about Simon Moon, Robert Putney Drake and Markoff Chaney are 99% me. Everything else is impossible to untangle. The celebrated Blow Job on the beach, for instance, is almost all Shea ... "
When asked whether the Lovecraft/Cthulhu elements in Illuminatus! come from Shea or Wilson, Wilson responded, " It’s me. I went through a period in the early 1960s when I kept having the Lovecraft horrors every time I took peyote."
As for Shea, he also says (in an interview for Outworlds), "The use made in Illuminatus! of Lovecraft’s material is largely Wilson’s contribution."
I don't know of a Shea interview that goes into detail on who wrote what, but this comment, also from the Outworlds interview details the process of cutting that Illuminatus! went through: "The original manuscript was about 1,200 typewritten pages, and we were asked to cut about 200 pages, which we did, screaming in agony all the while. This didn’t do any mortal, structural damage to the novel. It did cost us some good writing and some funny bits. A lot of what was cut was occult information in the appendices. Also When Atlantis Ruled the Earth was originally a complete screenplay-within-a-novel. Now it’s down to a summary."
CCN: Well, I want to touch upon your thoughts about the Internet a little bit later, but one thing I want to talk about since we mentioned Shea is that just to get into the mechanics of a writer, how did you collaborate?
RAW: Well, different writers have had different techniques. What Shea and I agreed on is to write alternate sections, and then I, how shall I phrase it? I persuaded Shea to let me rewrite his sections in order to make the style more uniform. So there are many sections that are almost all Shea in content, but they’re still me in style if you know what I mean. Like, one of the longest sections that’s almost all Shea is the movie about Atlantis… …yet the style is me. I rewrote the thing to get it into the style of the rest of the book, and I added a few key things like the idea of the fur bristling as an expression of emotion and a few other things like that. I also come up with the clouded lenses and I was trying to figure out how people who didn’t have our concepts of sin or mental illness would describe somebody whose perceptions they couldn’t understand, so I came up with the metaphor of the clouded lenses.
The interview also is interesting because of Wilson's comments on the Bride of Illuminatus sequel, which apparently didn't get very far: "Well, anyway, we decided we could do it, and we got started and then Shea died of cancer which was…the major tragedy of the last couple of years of my life. He was my best friend I think. So I’m finishing it on my own…."