A question of considerable interest to cell membrane biology is whether phase segregated domains across an asymmetric bilayer are strongly correlated with each other and whether phase segregation in one leaflet can induce segregation in the other. We answer both these questions in the affirmative, using an atomistic molecular dynamics simulation to study the equilibrium statistical properties of a 3-component asymmetric lipid bilayer comprising an unsaturated palmitoyl-oleoyl-phosphatidyl-choline, a saturated sphingomyelin, and cholesterol with different composition ratios. Our simulations are done by fixing the composition of the upper leaflet to be at the coexistence of the liquid ordered (l(o))-liquid disordered (l(d)) phases, while the composition of the lower leaflet is varied from the phase coexistence regime to the mixed l(d) phase, across a first-order phase boundary. In the regime of phase coexistence in each leaflet, we find strong transbilayer correlations of the l(o) domains across the two leaflets, resulting in bilayer registry. This transbilayer correlation depends sensitively upon the chain length of the participating lipids and possibly other features of lipid chemistry, such as degree of saturation. We find that the l(o) domains in the upper leaflet can induce phase segregation in the lower leaflet, when the latter is nominally in the mixed (l(d)) phase.