  We formulate integral Iwasawa main conjectures for suitable twists of a
newform $f$ that is non-ordinary at $p$, over the cyclotomic
$\mathbb{Z}_p$-extension, the anticyclotomic $\mathbb{Z}_p$-extensions (in both
the definite and the indefinite cases) as well as the
$\mathbb{Z}_p^2$-extension of an imaginary quadratic field $K$ where $p$
splits. In order to do so, we define Kobayashi-Sprung-style signed Coleman
maps, which we use to introduce doubly-signed Selmer groups. In the same
spirit, we construct signed (integral) Beilinson-Flach elements (out of the
collection of unbounded Beilinson-Flach elements of Loeffler-Zerbes), which we
use to define doubly-signed $p$-adic $L$-functions. The main conjecture then
relates these two set of objects. Furthermore, we show that the integral
Beilinson-Flach elements form a locally restricted Euler system, that in turn
allows us to deduce (under certain technical assumptions) one inclusion in each
one of the four main conjectures we formulate here (which may be turned into
equalities under favourable circumstances).
