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We use the coupled cluster method to study the zero-temperature quantum phasediagram of the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ model onthe honeycomb bilayer lattice. In each layer we include both nearest-neighborand frustrating next-nearest-neighbor antiferromagnetic exchange couplings, ofstrength $J_{1}>0$ and $J_{2} \equiv \kappa J_{1} > 0$, respectively. The twolayers are coupled by an interlayer nearest-neighbor exchange, with couplingconstant $J_{1}^{\perp} \equiv \delta J_{1}>0$. We calculate directly in theinfinite-lattice limit both the ground-state energy per spin and the N\'{e}elmagnetic order parameter, as well as the triplet spin gap. By implementing themethod to very high orders of approximation we obtain an accurate estimate forthe full boundary of the N\'{e}el phase in the $\kappa\delta$ plane. For eachvalue $\delta < \delta_{c}^{>}(0) \approx 1.70(5)$ we find an upper criticalvalue $\kappa_{c}(\delta)$, such that N\'{e}el order is present for $\kappa <\kappa_{c}(\delta)$. Conversely, for each value $\kappa < \kappa_{c}(0) \approx0.19(1)$ we find an upper critical value $\delta_{c}^{>}(\kappa)$, such thatN\'{e}el order persists for $0 < \delta < \delta_{c}^{>}(\kappa)$. Mostinterestingly, for values of $\kappa$ in the range $\kappa_{c}(0) < \kappa <\kappa^{>} \approx 0.215(2)$ we find a reentrant behavior such that N\'{e}elorder exists only in the range $\delta_{c}^{<}(\kappa) < \delta <\delta_{c}^{>}(\kappa)$, with $\delta_{c}^{<}(\kappa)>0$. These latter upperand lower critical values coalesce when $\kappa = \kappa^{>}$, such that$\delta_{c}^{<}(\kappa^{>}) = \delta_{c}^{>}(\kappa^{>}) \approx 0.25(5)$.

A frustrated honeycomb-bilayer Heisenberg antiferromagnet: The spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ model