I think the error is in Joyce's definition 4.12 of $\mathop{\mathrm{Spec}}\mathfrak{C}$! Consider the case when $\mathfrak{C}$ has no real points (example below): then for all $c \in \mathfrak{C}$, $U_c = \emptyset$ (as they are subsets of $X_\mathfrak{C} = \emptyset$), but definition 4.12 attempts to set $O_{X_\mathfrak{C}}(U_c) = \mathfrak{C}[c^{-1}]$ which is doubly wrong: (1) it depends on $c$, (2) should be $0$ if the construction were really a sheaf.
Notice that Joyce prefaces the definition with "As in Moerdijk, van Quê and Reyes [49, §1 & §3]" but I looked up that paper and did not find exactly this definition of $\mathop{\mathrm{Spec}}\mathfrak{C}$:
In §1 (as Joyce also explains), the authors don't deal with this version of the spectrum, but rather one based on all $C^\infty$-radical prime ideals and I think that in that setting, just as in algebraic geometry, the definition $O(U_c) = \mathfrak{C}[c^{-1}]$ is not problematic.
In §3 they do define another version of the spectrum due to Eduardo Dubuc and based on real points, but not through explicit formulas, instead giving a categorical argument for the existence of an adjoint to the functor of global sections defined on locally $C^\infty$-ringed spaces whose stalks have residue field $\mathbb{R}$.
So the problematic formula $O_{X_\mathfrak{C}}(U_c) = \mathfrak{C}[c^{-1}]$ does not actually appear in Moerdijk, van Quê and Reyes, nor, in Dubuc's $C^\infty$ Schemes, where the real-points spectrum was originally defined.
The construction in definition 4.12 can't be too far from correct (and is quite possibly completely correct in the special case of fair $C^\infty$-rings), but I don't know exactly how to fix it.
Example of a $C^\infty$-ring with no real points. Take $\mathfrak{C} = C^\infty(\mathbb{R})/I$ where $I$ is the ideal of $C^\infty$ functions $\mathbb{R} \to \mathbb{R}$ with compact support. Suppose there was a morphism of $C^\infty$-rings $f : \mathfrak{C} \to \mathbb{R}$, and consider the composite $f \circ p$ where $p : C^\infty(\mathbb{R}) \to \mathfrak{C}$ is the canonical projection. That is a real point of $C^\infty(\mathbb{R})$ and is thus given by evaluation at some real number $x$, but since functions of compact support can take arbitrary values at $x$, evaluation at $x$ cannot descend to the quotient $\mathfrak{C}$, a contradiction.
Since $X_{\mathfrak{C}}$ is empty, the only possible value for $\Gamma(\mathop{\mathrm{Spec}}\mathfrak{C})$ is the zero ring, which is what proposition 4.15 says should happen: this example is not fair and its fairification $R^{fa}_{fg}(\mathfrak{C})$ is indeed the zero ring.
