  The automorphism group of a map acts naturally on its flags (triples of
incident vertices, edges, and faces). An Archimedean map on the torus is called
almost regular if it has as few flag orbits as possible for its type; for
example, a map of type $(4.8^2)$ is called almost regular if it has exactly
three flag orbits. Given a map of a certain type, we will consider other more
symmetric maps that cover it. In this paper, we prove that each Archimedean
toroidal map has a unique minimal almost regular cover. By using the Gaussian
and Eisenstein integers, along with previous results regarding equivelar maps
on the torus, we construct these minimal almost regular covers explicitly.
