An incommensurate charge-density wave is a periodic modulation of charge that breaks translational symmetry at a momentum that does not coincide with the primitive lattice vectors. Its Goldstone excitation, the phason, comprises collective gapless phase fluctuations. Aiming to unveil the mechanism behind the onset of incommensurate charge order in layered materials, we study a half-filled, four-band tight-binding model on a ladder with a relative shift $\delta =p/q$ between the legs, induced by the dimerization of one of them. The shift results in a moiré supercell comprising $q$ composite cells and a modulated interleg tunneling. The moiré potential compresses the leg bands into flat minibands near the Fermi level, resulting in additional low-energy peaks in the density of states. Including Coulomb interactions, we find an incommensurate charge-density-wave phase in which the charge modulation is out of phase between the legs. The collective excitations of this state are long-lived neutral, acoustic phasons whose speed is controlled by the moiré parameter $\delta$ and the interleg tunneling amplitude. This model sheds light on the role of interlayer incongruities in the formation of excitonic charge-ordered phases in van der Waals and heterostructured materials.

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• Charge density waves
• Charge dynamics
• Density of states
• Flat bands
• Tight-binding model