  Let $E$ be an elliptic curve over an imaginary quadratic field $K$, and $p$
be an odd prime such that the residual representation $E[p]$ is reducible. The
$\mu$-invariant of the fine Selmer group of $E$ over the anticyclotomic
$\mathbb{Z}_p$-extension of $K$ is studied. We do not impose the Heegner
hypothesis on $E$, thus allowing certain primes of bad reduction to decompose
infinitely in the anticyclotomic $\mathbb{Z}_p$-extension. It is shown that the
fine $\mu$-invariant vanishes if certain explicit conditions are satisfied.
Further, a partial converse is proven.
