Apologies for the long question. Let $X=\mathbf P^1(\mathbf R) \subseteq \mathbf P^1(\mathbf C)$ be the real projective line. Let $\mathcal O_X$ be the sheaf of real-analytic complex-valued functions on $X$. For an open set $U\subseteq X$, $\mathcal O_X(U)$ is the $\mathbf C$-algebra of functions $U\to \mathbf C$ admitting a convergent power series expansion in the neighborhood of any $x \in U$.
I am wondering if there exists a nice algebraic descriptions of $R=\mathcal O_X(X)$, the ring of global analytic functions on $X$. This ring is interesting to me for a number of reasons. By Liouville's theorem, analytic functions on $\mathbf P^1(\mathbf C)$ are just constant functions. But on $X$, there are lots of nonconstant functions; $R$ contains things such as $x \mapsto \frac{x-i}{x+i}$, for instance. There are also many algebraic functions in $R$; for instance, $f(x)=\sqrt{\frac{1}{x^2+1}}$ lives in $R$. The ring $R$ has many desirable algebraic properties, which gives me hope that a purely algebraic description of it might exist.
(I made a few searches to see if I could find anything about this. Sadly, it seems that real-analytic manifolds are much less popular than their complex counterparts, and I couldn't find anything. If you know a good, short survey of the theory of real-analytic manifolds, I'd be grateful if you could point me to it.)
What I know:
There exists a natural injective map $R\to C^\infty(X)$, but it is far from being surjective. No good.
$R$ naturally contains the subring of the rational function field $\mathbf C(x)$ consisting of those rational functions having no poles along $X$.
Since $X$ is compact, $R$ has the sup norm, under which it is not complete. The subring of $\mathbf C(x)$ just described is dense in it.
$R$ is a UFD whose maximal ideals correspond naturally to the points of $X$. A version of the residue theorem holds on $X$. Functions which vanish nowhere are units.
Since every $f \in \mathbf C[[x]]$ which converges in an open interval around $0$ converges in an open disc of the same radius in $\mathbf C$, and since $X$ is compact, it follows that every $f \in R$ extends to a holomorphic function in a neighborhood of $X$ in $\mathbf P^1(\mathbf C)$. Conversely, every such holomorphic function determines an element of $R$. It follows that $R= \varinjlim \mathcal O_{\mathbf P^1(\mathbf C)}(V),$ the limit being taken over the neighborhoods $V$ of $X$ in $\mathbf P^1(\mathbf C)$. (In particular, $R$ contains functions which may have essential singularities away from the real line.)
From the previous point, it follows that the principle of analytic continuation holds in $R$. In particular, any $f\in R$ is determined by its power series expansion around any point (despite this power series not converging everywhere at once), and therefore, after any choice of point, $R$ embeds naturally in the subring $S$ of $\mathbf C[[x]]$ consisting of those power series which converge in a neighborhood of $0$. However, once again, $S$ is larger than $R$, because there are holomorphic functions on the open unit disc which do not extend outside of the disc.
The real projective line is isomorphic to the circle via stereographic projection, where the circle is given the structure of a real-analytic manifold by the fact that the transition maps of the usual charts $\theta \mapsto e^{i\theta}$ are analytic. By Fourier theory, every continuous function on the circle has a uniformly convergent expansion $\sum_{n \in \mathbf Z} a_n e^{in \theta}$. Therefore, the question can be reduced to finding those series $f(\theta)= \sum_{n \in \mathbf Z} a_n e^{in \theta}$ for which $\theta \mapsto f(\theta)$ is analytic in $\theta$. Perhaps a bound on the growth of the coefficients?