  Consider a pair consisting of an abstract tropical curve and an effective
divisor from the linear system associated to $k$ times the canonical divisor
for $k \in \bZ_{\geq 1}$. In this article we give a purely combinatorial
criterion to determine if such a pair arises as the tropicalization of a pair
consisting of a smooth algebraic curve over a non-Archimedean field with
algebraically closed residue field of characteristic~0 together with an
effective pluri-canonical divisor. To do so, we introduce tropical normalized
covers as special instances of tropical Hurwitz covers and reduce the
realizability problem for pluri-canonical divisors to the realizability problem
for normalized covers. Our main result generalizes the work of
M\"oller-Ulirsch-Werner on realizability of tropical canonical divisors and
incorporates the recent progress on compactifications of strata of
$k$-differentials in the work of Bainbridge-Chen-Gendron-Grushevsky-M\"oller.
