What is the Real Prime? There seems to be an importance to the ring of adeles for the rational numbers (discussed here), with valuations for every $\mathbb{Q}_p$, but also one "infinite" valuation "$\mathbb{Q}_∞$", seemingly equal to $\mathbb{R}$.
Why would something like $\mathbb{Q}_∞$ be used in the first place, and how is that equal to the reals? Is there something like a $∞$-adic metric that works like the usual one?
Moreover, it seems to suggest that $∞$ here is a sort of an infinite prime number, i.e. the real prime, having some occult-sounding books written about it. So, does it exist as some sort of a describable object here, or is it just notation?

It is a formal notation. We treat $|\cdot|$ as an absolute
value $|\cdot|_{\infty}$ coming from an “infinite prime”, so that we obtain, among other things, a product formula
$$
\prod_{p\le \infty} |\alpha|_p=1
$$
for every $\alpha\in \Bbb Q^{\times}$. Of course, $p=\infty$ is not really a prime. So $\Bbb Q_{\infty}:=\Bbb R$ is just a notation. We can summarize all completions of $\Bbb Q$ by
$$
\Bbb Q_2,\Bbb Q_3,\Bbb Q_5,\cdots ,\Bbb Q_{\infty}
$$
