Let
K
K
be an imaginary quadratic field and
p
p
be an odd prime which splits in
K
K
. Let
E
1
E_1
and
E
2
E_2
be elliptic curves over
K
K
such that the
Gal
(
K
¯
/
K
)
\operatorname {Gal}(\bar {K}/K)
-modules
E
1
[
p
]
E_1[p]
and
E
2
[
p
]
E_2[p]
are isomorphic. We show that under certain explicit additional conditions on
E
1
E_1
and
E
2
E_2
, the anticyclotomic
Z
p
\mathbb {Z}_p
-extension
K
anti
K_{\operatorname {anti}}
of
K
K
is integrally diophantine over
K
K
. When such conditions are satisfied, we deduce new cases of Hilbert’s tenth problem. In greater detail, the conditions imply that Hilbert’s tenth problem is unsolvable for all number fields that are contained in
K
anti
K_{\operatorname {anti}}
. We illustrate our results by constructing an explicit example for
p
=
3
p=3
and
K
=
Q
(
−
5
)
K=\mathbb {Q}(\sqrt {-5})
.