A supersolid combines density order with phase coherence, and doped lattice solids ask whether added defects can become coherent without melting the ordered background. We study a soft-core Bose-Hubbard model with isotropic hopping and an engineered non-axisymmetric dipolar interaction,
Vij=V2(xij2−yij2)/rij5+W6/rij6, where the sign-changing
dx2−y2 component selects a fixed
(q,0) stripe channel and the
W6/r6 core stabilizes the short-distance attractive branch. Using sign-problem-free quantum Monte Carlo method with worm algorithm, we find that the half-filled stripe parent responds asymmetrically to doping: the hole side forms locked commensurate stripe solids with vanishing superfluid stiffness, whereas the particle side forms a stripe supersolid with finite compressibility
κ>0, finite superfluid stiffness
ρs>0, and enhanced double occupancy
D. Keeping the same off-site kernel while increasing
U/t toward the hard-core limit shows that the particle-side supersolid disappears once doublon-like defects are projected out. Thus the engineered dipolar kernel selects the fixed
(q,0) stripe channel, while onsite softness selects the phase-coherent defect sector.
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 minimal hard-core Bose lattice model on a square optical lattice, with particles interacting through a sign-changing non-axisymmetric dipolar tail \mathcal V(\mathbf r)\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 stripe solid 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 with filling, showing that the robust ordered object is a family of stripe states rather than one fixed commensurate 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.