  We consider a homogenization problem associated with quasi-crystalline
multiple integrals of the form \begin{equation*} \begin{aligned}
u_\varepsilon\in L^p(\Omega;\mathbb{R}^d) \mapsto \int_\Omega
f_R\Big(x,\frac{x}{\varepsilon}, u_\varepsilon(x)\Big)\, dx, \end{aligned}
\end{equation*} where $u_\varepsilon$ is subject to constant-coefficient linear
partial differential constraints. The quasi-crystalline structure of the
underlying composite is encoded in the dependence on the second variable of the
Lagrangian, $f_R$, and is modeled via the cut-and-project scheme that
interprets the heterogeneous microstructure to be homogenized as an irrational
subspace of a higher-dimensional space. A key step in our analysis is the
characterization of the quasi-crystalline two-scale limits of sequences of the
vector fields $u_\varepsilon$ that are in the kernel of a given
constant-coefficient linear partial differential operator, $\mathcal{A}$, that
is, $\mathcal{A} u _\varepsilon =0$. Our results provide a generalization of
related ones in the literature concerning the ${\rm \mathcal{A} =curl } $ case
to more general differential operators $\mathcal{A}$ with constant
coefficients, and without coercivity assumptions on the Lagrangian $f_R$.
