= Solution
For each pair of inputs, define the joint probabilities
$$
\boxed{P(x,y|a,b)=\frac14[1+xyf(\theta(a,b))]},
\qquad x,y\in\{-1,1\}.
$$
They are nonnegative because $|f|\leq1$, sum to one, and have the required correlation:
$$
\sum_{x,y}xyP(x,y|a,b)=f(\theta(a,b)).
$$
Both marginals are uniform,
$$
\sum_yP(x,y|a,b)=\frac12,
\qquad
\sum_xP(x,y|a,b)=\frac12,
$$
independently of the remote input. The device is thus a <no-signalling box>: its superquantum correlation does not by itself transmit a message, so it is compatible with <relativistic causality>.
Solved by gpt-5.6-sol high.
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