  We study the ground-state phase diagram of a spin-$\frac{1}{2}$ frustrated
XXZ ladder, in which two antiferromagnetic chains are coupled by competing rung
and diagonal interactions, $J_\perp$ and $J_\times$. Previous studies on the
isotropic model have revealed that a fluctuation-induced effective dimer
attraction between the legs stabilizes the columnar dimer (CD) phase in the
highly frustrated regime $J_\perp\approx 2J_\times$, especially for
ferromagnetic $J_{\perp, \times}<0$. By means of effective field theory and
numerical analyses, we extend this analysis to the XXZ model, and obtain a rich
phase diagram. The diagram includes four gapped featureless phases with no
symmetry breaking: the rung singlet (RS) and Haldane phases as well as their
twisted variants, the RS* and Haldane* phases, which are all distinct in the
presence of certain symmetries. Significantly, the Haldane-CD transition point
in the isotropic model turns out to be a crossing point of two transition lines
in the XXZ model, and the stripe N\'eel and RS* phases appear between these
lines. This indicates a nontrivial interplay between the effective dimer
attraction and the exchange anisotropy. In the easy-plane regime, the four
featureless phases and two critical phases are found to compete in a complex
manner depending on the signs of $J_{\perp, \times}$.
