Why is it called "The Real Bitcoin"?
Bitcoin has many important properties but by far Bitcoin's most important property is its immutability. Without immutability all other properties—the 21 million supply cap, the mining reward schedule, the ability to send any amount of coins to any address by signing with the correct private key, censorship and seizure resistance, etc.—become open to interpretation and political meddling.
One of Bitcoin's primary uses is that of a non-inflationary store of value. People who understand this use Bitcoin to save money for the long term. The really, really long term. Imagine if Bitcoin was like any preexisting centralized service, or any number of so-called "alt coins", where monetary policy can be changed at the whims of some group of individuals. How could you be sure then that the bitcoin you have saved wouldn't somehow change in the future? The answer is you couldn't, and that's what separates Bitcoin from all other tokens. It is completely incorruptible.
So if what it means to hold bitcoin now is that you can rest easy knowing that no matter what happens—in five years, in ten years, in 100 years—its properties will not change, then that same level of assurance must apply to holders of bitcoin at now - N (where N < the age of Bitcoin). That is to say, past hodlers. This is where The Real Bitcoin (TRB) comes in. TRB adheres to the original Satoshi consensus rules, without any added layers (enforced by miners or otherwise). TRB works only with traditional Bitcoin addresses, those starting with a 1. It does not enforce SegWit rules and in fact sees SegWit addresses as "anyone-can-spend" addresses. In short, TRB is Bitcoin as it was from 2009 to 2017, before Gavin, Hearn, Ver, and other big block sabetuers started stirring up controversy which culminated in the "segregated witness" soft fork (and of course the BCH lol-fork, but that was largely just a sideshow to make SegWit appear necessary and—compared to "The New York Agreement" (SegWit2x)—the more palatable choice).
TRB is the only remaining actively maintained Bitcoin client that respects the original consensus rules—without compromise and without qualifications. For these reasons, it is The Real Bitcoin.
How to Install TRB
TRB is not for casual users, however it is also really not that difficult to stand up and maintain if you know your way around Linux. There are currently three build methods available. They all result in the same TRB client so the choice depends largely on personal preference and host environment.
Build From Source Via the Original 'Rotor' Method
This method was developed by asciilifeform (WOT: asciilifeform) and is the only method that produces a deterministicly built static binary. Provided you are working on a machine with a version of glibc < 2.28 this method is fairly straightfoward. Because the resulting binary is statically linked you can throw it on a thumb drive and run it on any other Linux system you have lying around. This method can also be used on a glibc-based system where the glibc version is 2.28 or greater by creating a Dulap chroot, building from inside the chroot, and then copying the resulting binary over to your host system. Instructions for the complete build process can be found here.
Build From Source Via the System Compiler
This method was developed by jfw (WOT: jfw) and works well on musl-based systems. Depending on the versions of various build dependencies on your system you may have to roll your sleeves up and make some modifications. The complete process, including some known obstacles and their workarounds, is outlined here. Additional details can be found here.
Build From Source Via the 'Fetch-Bitcoind' Build Script
This method, also developed by jfw, is an attempt to streamline the process by automating the step of collecting the various patches and signatures required to press the TRB source. I have not yet personally tested this method but since I had good results with jfw's 'system compiler' method I don't mind recommending it here for others. Details and instructions can be found here.