Microwave-dressed polar molecules offer a route to lattice quantum simulators in which the angular form of long-range dipolar interactions, not only their overall strength, can be engineered. We study this setting in a hard-core Bose lattice model on a square optical lattice, with particles interacting through a sign-changing nonaxisymmetric dipolar tail $\mathcal{V}\left(\mathbf{r}\right)\propto (x^2-y^2)/{(x^2+y^2)}^{5/2}$ that is repulsive along one lattice axis and attractive along the other. Using worm-algorithm path-integral quantum Monte Carlo simulations, supported by a hard-core spin mapping and a Gutzwiller soft-mode diagnostic, we find two regimes controlled by $t/V$: At larger $t/V$ the system remains superfluid but develops a pronounced directional stiffness anisotropy, while at smaller $t/V$ it forms a sequence of commensurate stripe solids selected in the $(q,0)$ axial family, corresponding to real-space stripes parallel to $y$. The leading-ordering wave vector remains in this axial family but reorganizes between commensurate filling plateaus, showing that the robust ordered object is a family of stripe states rather than one fixed Bragg peak. Near the closure of the stripe lobe, averaged observables can mimic a narrow supersolid signal; measurement-resolved stripe structure-factor histograms instead reveal first-order switching between superfluid and stripe-solid sectors. NaCs lattice estimates place the relevant $V/t$ window within reach of modest effective dressed dipole moments, linking the predicted stripe-family order and its experimental diagnostics to accessible molecular quantum-simulation scales.

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• Bose gases
• Cold gases in optical lattices
• Dipolar gases
• Dipolar interaction
• Dipolar molecules
• Quantum criticality
• Superfluids
• Ultracold gases
• Monte Carlo methods