  Let $K$ be a imaginary quadratic field where the prime $p$ splits. Our goal
in this article is to prove results towards the Iwasawa main conjectures for
$p$-nearly-ordinary families associated to $\mathrm{GL}_2\times
\mathrm{Res}_{K/\mathbb{Q}}\mathrm{GL}_1$ with a minimal set of assumptions.
The main technical input is an improvement on the locally restricted Euler
system machinery that allows the treatment of residually reducible cases, which
we apply with the Beilinson--Flach Euler system.
