  Dressed Rydberg atoms in optical lattices are a promising platform for the
quantum simulation of intriguing phenomena emerging in strongly interacting
systems. Relevant to such a setup, we investigate the phase diagram of
hard-core bosons in a triangular ladder with next-to-nearest-neighbor
interaction along each leg and nearest-neighbors interactions without hopping
between the legs. For weak interactions, Abelian bosonization predicts a spin
density wave and a fully gapless Luttinger liquid phase. Such liquids
transition to a 'spin-locked' cluster Luttinger liquid at strong interactions
along each leg, as predicted by cluster bosonization. Interestingly, the
competition with the zigzag interaction generates a charge density wave, a
'polarized holonic' phase, and a crystalline phase at the filling 2/5, that we
address via semi-classical perturbative approach. Exact diagonalization and
density matrix renormalization group simulations confirm the predictions and
further characterize the phases and their transitions.
