It is well known that the formation and spatial correlation of lipid domains in the two apposed leaflets of a bilayer are influenced by weak lipid-lipid interactions across the bilayer's midplane. Transmembrane proteins span through both leaflets and thus offer an alternative domain coupling mechanism. Using a mean-field approximation of a simple bilayer-type lattice model, with two two-dimensional lattices stacked one on top of the other, we explore the role of this "structural" inter-leaflet coupling for the ability of a lipid membrane to phase separate and form spatially correlated domains. We present calculated phase diagrams for various effective lipid-lipid and lipid-protein interaction strengths in membranes that contain a binary lipid mixture in each leaflet plus a small amount of added transmembrane proteins. The influence of the transmembrane nature of the proteins is assessed by a comparison with "peripheral" proteins, which result from the separation of one single integral protein into two independent units that are no longer structurally connected across the bilayer. We demonstrate that the ability of membrane-spanning proteins to facilitate domain formation requires sufficiently strong lipid-protein interactions. Weak lipid-protein interactions generally tend to inhibit phase separation in a similar manner for transmembrane as for peripheral proteins.