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Type Ia supernovae are triggered by accretion onto a white dwarf from a companion which is most likely Roche lobe-filling at the time of the explosion. The collision between the ejecta and a surviving companion carves out a conical wake, which could manifest as an asymmetry when the ejecta reaches the remnant phase. We simulate the companion interaction using the Athena++ hydrodynamics solver to determine the ejecta structure for a double-degenerate type Ia supernova. Ejecta in the wake is of lower density and higher velocity than the unperturbed ejecta. We then evolve the ejecta for several thousand years using the expanding-grid code Sprout. The forward shock within the wake is initially indented, but becomes spherical after roughly a thousand years due to transverse motion of shocked ejecta that fills the wake. The reverse shock travels quickly within the wake, leading to an off-center convergence of the reverse shock and leaving the remnant with an asymmetrical core. This also draws material from the interstellar medium deep into the remnant, eventually reaching the center. Large Rayleigh-Taylor plumes are found around the edge of the wake, creating a toroidal structure composed primarily of ejecta. Estimates of the thermal X-ray emission show that such remnants exhibit observable asymmetries for thousands of years.