  We define a Mendelsohn triple system (MTS) with self-distributive quasigroup
multiplication and order coprime with $3$ to be distributive, non-ramified
(DNR). We classify, up to isomorphism, all DNR MTS and enumerate isomorphism
classes (extending the work of Donovan, Griggs, McCourt, Opr\v{s}al, and
Stanovsk\'{y}). The classification is accomplished via the representation
theory of the Eisenstein integers, $\mathbb{Z}[\zeta]=\mathbb{Z}[X]/(X^2-X+1)$.
Containing the class of DNR MTS is that of MTS with an entropic (linear over an
abelian group) quasigroup operation. Partial results on the classification of
entropic MTS with order divisible by $3$ are given, and a complete
classification is conjectured. We also prove that for any entropic MTS, the
qualities of being non-ramified, pure, and self-orthogonal are equivalent. We
introduce the varieties $\mathbf{RE}$ and $\mathbf{LE}$ of (resp. right and
left) Eisenstein quasigroups, whose respective linear representation theories
correspond to the alternative presentation $\mathbb{Z}[X]/(X^2+X+1)$ of the
Eisenstein integers.
