  We investigate the behavior of the frustrated $J_1$-$J_2$ Ising model on a
square lattice under the influence of random dilution and spatial anisotropies.
Spinless impurities generate a random-field type disorder for the spin-density
wave (stripe) order parameter. These random fields destroy the long-range
stripe order in the case of spatially isotropic interactions. Combining
symmetry arguments, percolation theory and large-scale Monte Carlo simulations,
we demonstrate that arbitrarily weak spatial interaction anisotropies restore
the stripe phase. More specifically, the transition temperature $T_c$ into the
stripe phase depends on the interaction anisotropy $\Delta J$ via $T_c \sim
1/|\ln (\Delta J)|$ for small $\Delta J$. This logarithmic dependence implies
that very weak anisotropies are sufficient to restore the transition
temperature to values comparable to that of the undiluted system. We analyze
the critical behavior of the emerging transition and find it to belong to the
disordered two-dimensional Ising universality class, which features the clean
Ising critical exponents and universal logarithmic corrections. We also discuss
the generality of our results and their consequences for experiments.
