The programmable self-assembly of nano-scale and micro-scale components into appropriate, desired structures by specific binding of complementary DNA sequences is an auspicious route to fabricating novel, functional materials. This approach can be employed for lipid vesicles by utilizing single-stranded DNA (ssDNA) with hydrophobic modifications that anchor the ssDNA to the lipid membrane. By exploiting such molecules, reversible assembly of multi-compartment vesicle super-structures has been demonstrated for vesicles ranging from 100 nm to several microns in diameter. The variety of different lipid components that are readily available makes it possible to tune the properties and interactions of the vesicles in these systems not only by choice of DNA sequences, but also by choice of the lipid composition of the vesicles. Here we demonstrate that lateral phase separation in the lipid membrane can be used to break the spherical symmetry of the interaction potential between DNA-modified vesicles. We examine how membrane-anchored DNA partitions between membranes phases in regimes of fluid-solid coexistence (where the solid phase is either L_β or P_β') and liquid-liquid coexistence. We investigate history-dependent and composition-dependent effects on DNA partitioning as well as demonstrate that the DNA-partitioning is dependent on the chosen method of membrane anchoring. This technique of laterally separating DNA-bound species on a vesicle could be used to engineer functional surface domains for technological applications.