  We study $k$-tilings ($k$-tuples of domino tilings) of the Aztec diamond of
rank $m$. We assign a weight to each $k$-tiling, depending on the number of
dominos of certain types and the number of "interactions" between the tilings.
Employing the colored vertex models introduced in earlier work to study
supersymmetric LLT polynomials, we compute the generating polynomials of the
$k$-tilings. We then prove some combinatorial results about $k$-tilings,
including a bijection between $k$-tilings with no interactions and $1$-tilings,
and we compute the arctic curves of the tilings for $t=0$ and
$t\rightarrow\infty$. We also present some lozenge $k$-tilings of the hexagon
and compute the arctic curves of the tilings for $t=0$.
