  We find that the phase appearing in the unitarity relation between
$\mathcal{B}(K_L\rightarrow \mu^+\mu^-)$ and $\mathcal{B}(K_L\rightarrow
\gamma\gamma)$ is equal to the phase shift in the interference term of the
time-dependent $K\rightarrow \mu^+\mu^-$ decay. A probe of this relation at
future kaon facilities constitutes a Standard Model test with a theory
precision of about $2\%$. The phase has further importance for sensitivity
studies regarding the measurement of the time-dependent $K\rightarrow
\mu^+\mu^-$ decay rate to extract the CKM matrix element combination $\vert
V_{ts} V_{td} \sin(\beta+\beta_s)\vert\approx A^2\lambda^5\bar\eta$. We find a
model-independent theoretically clean prediction, $\cos^2\varphi_0 = 0.96 \pm
0.03$. The quoted error is a combination of the theoretical and experimental
errors, and both of them are expected to shrink in the future. Using input from
the large-$N_C$ limit within chiral perturbation theory, we find a theory
preference towards solutions with negative $\cos\varphi_0$, reducing a
four-fold ambiguity in the angle $\varphi_0$ to a two-fold one.
