  Let $U$ be a smooth affine curve over a number field $K$ with a
compactification $X$ and let $\mathbb L$ be a rank $2$, geometrically
irreducible $\bar{\mathbb Q}_\ell$-local system on $U$ with cyclotomic
determinant that extends to an integral model, has Frobenius traces all in some
fixed number field $E\subset \bar{\mathbb Q}_\ell$, and has bad, infinite
reduction at some closed point $x$ of $X\setminus U$. We show that $\mathbb L$
occurs as a summand of the cohomology of a family of abelian varieties over
$U$. The argument follows the structure of the proof of a recent theorem of
Snowden-Tsimerman, who show that when $E=\mathbb Q$, then $\mathbb L$ is
isomorphic to the cohomology of an elliptic curve $E_U\rightarrow U$.
