Consistency between the exchange-correlation (XC) functional used during pseudopotential construction and planewave-based electronic structure calculations is important for an accurate and reliable description of the structure and properties of condensed-phase systems. In this work, we present a general scheme for constructing pseudopotentials with range-separated hybrid (RSH) XC functionals based on the solution of the all-electron radial integro-differential equation for a spherically symmetrized reference atomic configuration. As a proof of principle, we demonstrate pseudopotential construction with the Perdew-Burke-Ernzerhof (PBE), hybrid PBE (PBE0), Heyd-Scuseria-Ernzerhof RSH (HSE06), and screened RSH (SRSH, based on the long-range corrected LC- RSH) XC functionals for a select set of atoms and then investigate the importance of pseudopotential consistency when computing band gaps, equilibrium lattice parameters, bulk moduli, and atomization energies of several solid-state systems. In doing so, we find that pseudopotential consistency errors tend to be systematic and can be as large as 0.1 eV (or ) when computing band gaps.