  For a positive proportion of primes $p$ and $q$, we prove that $\mathbb{Z}$
is Diophantine in the ring of integers of $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$.
This provides a new and explicit infinite family of number fields $K$ such that
Hilbert's tenth problem for $O_K$ is unsolvable. Our methods use Iwasawa theory
and congruences of Heegner points in order to obtain suitable rank stability
properties for elliptic curves.
