= Solution
The <EPR criterion of reality> proposes that a quantity has an element of physical reality if its value can be predicted with certainty without disturbing the system. To use remote predictions this way, one additionally assumes locality: the choice and execution of a <quantum measurement> on separated particles cannot disturb the local physical properties of the remaining particle. The contradiction below is with this combination of the criterion and locality, rather than with a definition of reality alone.
Use the <computational basis> $|0\rangle=|\uparrow\rangle$, $|1\rangle=|\downarrow\rangle$ and the conventional <Pauli matrices>
$$
X|0\rangle=|1\rangle,\quad X|1\rangle=|0\rangle,\qquad
Y|0\rangle=i|1\rangle,\quad Y|1\rangle=-i|0\rangle.
$$
For each site $j$, let $O_j$ be the <tensor product> of $X$ at that site and $Y$ at all other sites. For $N=4M+3$, the number of $Y$ factors is $N-1=4M+2$, so
$$
O_j|0^N\rangle=i^{N-1}|1^N\rangle=-|1^N\rangle,\qquad
O_j|1^N\rangle=(-i)^{N-1}|0^N\rangle=-|0^N\rangle.
$$
The minus sign in the given <GHZ state> then implies
$$
O_j|\psi\rangle=|\psi\rangle\quad\text{for every }j,
\qquad X^{\otimes N}|\psi\rangle=-|\psi\rangle.
$$
Hence \b[every one-$X$, remaining-$Y$ product has measurement outcome $+1$ with certainty, while the all-$X$ product has outcome $-1$ with certainty]. These are experimentally separate local <Pauli measurements> on identically prepared copies, not joint measurements of $X$ and $Y$ on the same particle.
For a chosen site, the other $N-1$ sites can determine its $X$ outcome with certainty by measuring $Y$ and using the corresponding $O_j$ relation. They can determine its $Y$ outcome with certainty by selecting an $O_\ell$ with $\ell\ne j$, measuring $X$ at $\ell$ and $Y$ at all other remote sites. Since $N\geq3$, both choices are available. Locality and the <EPR criterion of reality> therefore assign each site definite values $x_j,y_j\in\{+1,-1\}$, independent of which remote choice is made, even though <quantum theory> does not jointly measure the local $X$ and $Y$.
The certain one-$X$ correlations require those assigned values to obey
$$
x_j\prod_{k\ne j}y_k=+1\qquad(j=1,\ldots,N).
$$
Multiplying these scalar equations gives
$$
1=\prod_{j=1}^N x_j\prod_{k=1}^N y_k^{N-1}=\prod_{j=1}^N x_j,
$$
since $N-1$ is even and $y_k^2=1$. Thus the all-$X$ product would necessarily be $+1$. But its quantum value is $-1$, giving
$$
\boxed{\prod_jx_j=+1\ \text{from local predetermined values},\qquad
\prod_jx_j=-1\ \text{from quantum theory}.}
$$
The equations cannot hold even for one assignment of values, so allowing a probability distribution over assignments cannot remove the contradiction. This is the <GHZ contradiction for N congruent to 3 modulo 4> for every $M\geq0$. Multiplying classical assigned values is crucial: one must not commute the incompatible local <Pauli operators> as though they were ordinary numbers. Their noncommutativity is exactly what permits the quantum correlations to evade the classical product argument.
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