  We report the manifestation of field-induced Berezinskii-Kosterlitz-Thouless
(BKT) correlations in the weakly coupled spin-1/2 Heisenberg layers of the
molecular-based bulk material [Cu(pz)$_2$(2-HOpy)$_2$](PF$_6$)$_2$. Due to the
moderate intralayer exchange coupling of $J/k_\mathrm{B} = 6.8$ K, the
application of laboratory magnetic fields induces a substantial $XY$ anisotropy
of the spin correlations. Crucially, this provides a significant BKT regime, as
the tiny interlayer exchange $J^\prime / k_\mathrm{B} \approx 1$ mK only
induces 3D correlations upon close approach to the BKT transition with its
exponential growth in the spin-correlation length. We employ nuclear magnetic
resonance and $\mu^{+}$SR measurements to probe the spin correlations that
determine the critical temperatures of the BKT transition as well as that of
the onset of long-range order. Further, we perform stochastic series expansion
quantum Monte Carlo simulations based on the experimentally determined model
parameters. Finite-size scaling of the in-plane spin stiffness yields excellent
agreement of critical temperatures between theory and experiment, providing
clear evidence that the nonmonotonic magnetic phase diagram of
[Cu(pz)$_2$(2-HOpy)$_2$](PF$_6$)$_2$ is determined by the field-tuned $XY$
anisotropy and the concomitant BKT physics.
