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Abstract The large-scale morphology of Milky Way (MW)–mass dark matter halos is shaped by two key processes: filamentary accretion from the cosmic web and interactions with massive satellites. Disentangling their contributions is essential for understanding galaxy evolution and constructing accurate mass models of the MW. We analyze the time-dependent structure of MW-mass halos from zoomed cosmological-hydrodynamical simulations by decomposing their mass distribution into spherical harmonic expansions. We find that the dipole and quadrupole moments dominate the gravitational power spectrum, encoding key information about the halo’s shape and its interaction with the cosmic environment. While the dipole reflects transient perturbations from infalling satellites and damps on dynamical timescales, the quadrupole—linked to the halo’s triaxiality—is a persistent feature. We show that the quadrupole’s orientation aligns with the largest filaments, imprinting a long-lived memory on the halo’s morphology even in its inner regions (∼30 kpc). At the virial radius, the quadrupole distortion can reach 1–2 times the spherical density, highlighting the importance of environment in shaping MW-mass halos. Using multichannel singular spectrum analysis, we successfully disentangle the effects of satellite mergers and filamentary accretion on quadrupole. We find that, compared to isolated MW–LMC simulations that typically use a spherical halo, the LMC-mass satellite induces a quadrupolar response that is an order of magnitude larger in our cosmological halo. This highlights the need for models that incorporate the MW’s asymmetry and time evolution, with direct consequences for observable structures such as disk warps, the LMC-induced wake, and stellar tracers—particularly in the era of precision astrometry.