Multicomponent quantum chemical methods seek to include nuclear quantum effects of select nuclei in quantum chemistry calculations by not invoking the Born-Oppenheimer approximation for these nuclei. In multicomponent methods, the inclusion of electron-proton correlation is essential for obtaining even qualitatively accurate protonic densities. However, most of the recently developed multicomponent methods have either used or obtained molecular orbitals from a single-reference mean-field wave function that neglects all electron-proton correlation that is analogous to using Hartree-Fock orbitals in a single-component framework. We examine the consequences of using Hartree-Fock orbitals in multicomponent calculations by developing the multicomponent heat-bath configuration interaction (HCI) method. Multicomponent HCI is a multicomponent selected configuration interaction (CI) technique that enables an accurate approximation of a complete active space or truncated CI wave function for systems with large active spaces. The multicomponent HCI method is shown to reproduce the ground-state protonic density of the HeHHe, HCN, and FHF systems when compared to reference grid-based calculations. For all three systems, the coefficient of the leading configuration in the wave function expansion is less than 0.95, indicating that all systems have multireference character. This is highly noteworthy as none of the systems have multireference character in a single-component framework and suggests that multireference character appears inherent to or at least more commonly in a multicomponent framework than a single-component framework. Even when natural orbitals are used rather than Hartree-Fock orbitals for the multicomponent HCI calculations, aspects of the multireference character remain for FHF and HCN. Consequences and implications of the multireference character of multicomponent quantum chemical systems are discussed.