Multilib-LFS is a variant of the LFS book. This variant builds a system which is able to execute 32-bit binaries.
That sounds trivial, but it isn't. Nowadays, nearly all CPUs in our computers are 64-bit, and most of them are Intel-compatible. This has happened before, many years ago, where 16-bit CPUs (8086) were common and new 32-bit CPUs were coming up (80386). Common to all these CPUs is that they are able to execute machine instructions of previous architectures, meaning that a 32-bit CPU can execute 16-bit instructions, a 64-bit CPU can execute 32-bit instructions, and so on. The problem is that the binary format in which those instructions are stored differs.
Since this CPU architecture has existed for decades now, there is also a lot of software written for it. Unfortunatly, some of this software is only available in a pre-compiled binary, and no source code is available anymore. This can be due to several different reasons, such as the source code getting lost or never being published due to licensing issues or because the creator did not want to release the source code. In these cases though, the binary is still available.
Since the format of 64-bit and 32-bit binaries are different, these old programs cannot be executed on modern systems. This is where the multilib systems come to play and therefore Multilib-LFS. Multilib systems provide the environment so that 32-bit binaries can be executed on a 64-bit OS as well. This needs to be support in the kernel, but it also needs several other core libraries to be available in 32-bit versions.
But thats not all. There is another architecture which is somewhat between 32- and 64-bit. Its called x32. That is, the binaries operate with the full instruction set of a 64-bit CPU, but address space is limited to 32-bit (~4GB). Executing such binaries does either need a system built explicitly for x32 or a multilib system which is capable of running those binaries.
The Multilib-LFS book is based on the actual LFS book but has some extensions in the instructions. In most cases, this extensions are repetitions of the build instructions, but just in another mode (m32 and/or mx32).
Usually, there is no need to run a multilib system. But there might be circumstances where you need one:
There might be software you want to use where that software is 32bit and there are no sources available to recompile it. Even when source code is available though, it might not be possible to recompile because of unresolvable dependencies to other (old) packages. A printer driver only available as 32bit binary was one of the reasons to kick-off the mlfs project.
For virtualization (here VirtualBox) m32-multilib is named as a prerequisite. When you want to install VirtualBox on a LFS system to be your VM host, only four additional non-BLFS packages are need. If you want to rebuild the 64bit and 32bit guest additions, a multilib OS is required to build them.
Also, GLFS (Gaming LFS) depends on multilib.
There are three versions of the Multilib-LFS online:
MLFS is hard to test unless there are old binaries available to use. When such a binary is executable, it unfortunatly does not mean that any other of those binaries will run, too. There might be BLFS library packages required as dependencies in that particular multilib environment. Neither MLFS nor BLFS covers that at the moment (there is no MBLFS) but MLFS should give the user an idea on how to build multilib versions of libraries.