This question is the outcome of a few naive thoughts, without reading the proof of Gelfand-Neumark theorem.

Given a compact Hausdorff space $X$, the algebra of complex continuous functions on it is enough to capture everything on its space. In fact, by the Gelfand-Neumark theorem, it is enough to consider the commutative C*-algebras instead of considering compact Hausdorff spaces.

The important thing here is that $C*$-algebras have a *complex structure*. The real structure is not enough. Given the algebra $C(X, \mathbb R) \oplus C(X, \mathbb R)$ of real continuous functions on $X$, the algebra $C(X, \mathbb{C})$ is simply the direct sum $C(X, \mathbb R) \oplus C(X, \mathbb R)$, as a Banach algebra(and this can be given a complex structure, (seeing it as the complexification...)). But to obtain a C*-algebra, we need an additional C*-algebra, and the obvious way, ie, defining $(f + ig)$* $= (f - ig)$ does not work out. More precisely, the C* identity does not hold.

So one cannot weaken (as it stands) the condition in the Gelfand-Neumark theorem that we need the algebra of *complex* continuous functions on the space $X$, since we do need the C* structure. Of course, this is without an explicit counterexample. Which brings us to:

Qn 1. Please given an example of two non-homeomorphic compact Hausdorff spaces $X$ and $Y$ such that the function algebras $C(X, \mathbb R)$ and $C(Y, \mathbb R)$ are isomorphic(as real Banach algebras)?

(Here I am hoping that such an example exists).

Then again,

Qn 2. From the above it appears that the structure of complex numbers is involved when the algebra of complex functions captures the topology on the space. So how exactly is this happening?

(The vague notions concerning this are something like: the complex plane minus a point contains nontrivial $1$-cycles, so perhaps the continuous maps to the complex plane might perhaps capture all the information in the first homology, etc..)..

**Note** : Edited in response to the answers. Fixed the concerns of Andrew Stacey, and changed Gelfand-Naimark to Gelfand-Neumark, as suggested by Dmitri Pavlov.