We investigate dynamical and stationary compositional and surface morphologies in macroscopically phase-separating multicomponent lipid bilayer membranes using a computational model. We employ a phase-field method for the description of the coexisting phases and treat the two leaflets individually while including interleaflet interactions. The compositional evolution of the two leaflets is coupled to the shape evolution of the membrane via a Helfrich free energy with a composition-dependent spontaneous curvature. We investigate the effects of the interleaflet interaction on the dynamics and stationary states of a system favoring nonzero spontaneous curvatures. Morphological phase diagrams are mapped in composition space using three different interleaflet coupling strengths. We find that characteristics sensitive to the coupling strength include the time required to develop regions of fully separated phases, the prevalence of a stripe morphology, and the shifting of phase compositions to accommodate energetically favorable interactions across leaflets. Characteristics found to be robust with respect to coupling strength include (1) the stripe morphology is favored at nearly equal mixtures and (2) phase separation is prevented in systems where a pair of phases that preferentially interact across leaflets together occupy nearly all or none of the membrane.