In what sense is the IO Monad pure?

Also, isn't it possible to describe pretty much any non-pure function like a function of the real world? For example, can't we think of, say, C's malloc as being a function that takes a RealWorld and an Int and returns a pointer and a RealWorld, only just like in the IO monad the RealWorld is implicit?

For sure ...

The whole idea of functional programming is to describe programs as a combination of small, independent calculations building up bigger computations.

Having these independent calculations, you'll have lots of benefits, reaching from concise programs to efficient and efficiently parallelizable codes, laziness up to the the rigorous guarantee that control flows as intended - with no chance of interference or corruption of arbitrary data.

Now - in some cases (like IO), we need impure code. Calculations involving such operations cannot be independent - they could mutate arbitrary data of another computation.

The point is - Haskell is always pure, IO doesn't change this.

So, our impure, non-independent codes have to get a common dependency - we have to pass a RealWorld. So whatever stateful computation we want to run, we have to pass this RealWorld thing to apply our changes to - and whatever other stateful computation wants to see or make changes has to know the RealWorld too.

Whether this is done explicitly or implicitly through the IO monad is irrelevant. You build up a Haskell program as a giant computation that transforms data, and one part of this data is the RealWorld.

Once the initial main :: IO () gets called when your program is run with the current real world as a parameter, this real world gets carried through all impure calculations involved, just as data would in a State. That's what monadic >>= (bind) takes care of.

And where the RealWorld doesn't get (as in pure computations or without any >>=-ing to main), there is no chance of doing anything with it. And where it does get, that happened by purely functional passing of an (implicit) parameter. That's why

let foo = putStrLn "AAARGH" in 42

does absolutely nothing - and why the IO monad - like anything else - is pure. What happens inside this code can of course be impure, but it's all caught inside, with no chance of interfering with non-connected computations.

I think the best explanation I've heard was actually fairly recently on SO. IO Foo is a recipe for creating a Foo. Another common, more literal, way of saying this is that it is a "program that produces a Foo". It can be executed (many times) to create a Foo or die trying. The execution of the recipe/program is what we ultimately want (otherwise, why write one?), but the thing that is represented by an IO action in our code is the recipe itself.

That recipe is a pure value, in the same exact sense that a String is a pure value. Recipes can be combined and manipulated in interesting, sometimes astonishing, ways, but the many ways these recipes can be combined (except for the blatantly non-pure unsafePerformIO, unsafeCoerce, etc.) are all completely referentially transparent, deterministic, and all that nice stuff. The resulting recipe depends in absolutely no way whatsoever on the state of anything other than the recipes that it was built up from.