  We deal with the following system of coupled asymmetric oscillators \[
\begin{cases} \ddot{x}_1+a_1x_1^+-b_1x^-_1+\phi_1(x_2)=p_1(t) \\
\ddot{x}_2+a_2\,x_2^+-b_2\,x^-_2+\phi_2(x_1)=p_2(t) \end{cases} \] where
$\phi_i: \mathbb{R} \to \mathbb{R}$ is locally Lipschitz continuous and
bounded, $p_i: \mathbb{R} \to \mathbb{R}$ is continuous and $2\pi$-periodic and
the positive real numbers $a_i, b_i$ satisfy $$
\dfrac{1}{\sqrt{a_i}}+\dfrac{1}{\sqrt{b_i}}=\dfrac{2}{n}, \quad \mbox{ for some
} n \in \mathbb{N}. $$ We define a suitable function $L: \mathbb{T}^2 \to
\mathbb{R}^2$, appearing as the higher-dimensional generalization of the well
known resonance function used in the scalar setting, and we show how unbounded
solutions to the system can be constructed whenever $L$ has zeros with a
special structure. The proof relies on a careful investigation of the dynamics
of the associated (four-dimensional) Poincar\'e map, in action-angle
coordinates.
