  Mutual Coupling (MC) emerges as an inherent feature in Reconfigurable
Intelligent Surfaces (RISs), particularly, when they are fabricated with
sub-wavelength inter-element spacing. Hence, any physically-consistent model of
the RIS operation needs to accurately describe MC-induced effects. In addition,
the design of the ElectroMagnetic (EM) transmit/receive radiation patterns
constitutes another critical factor for efficient RIS operation. The latter two
factors lead naturally to the emergence of non-local RIS structures, whose
operation can be effectively described via non-diagonal phase shift matrices.
In this paper, we focus on jointly optimizing MC and the radiation patterns in
multi-user MIMO communication systems assisted by non-local RISs, which are
modeled via the scattering parameters. We particularly present a novel problem
formulation for the joint optimization of MC, radiation patterns, and the
active and passive beamforming in a physically-consistent manner, considering
either reflective or transmissive RIS setups. Differently from the current
approaches that design the former two parameters on the fly, we present an
offline optimization method which is solved for both considered RIS
functionalities. Our extensive simulation results, using both parametric and
geometric channel models, showcase the validity of the proposed optimization
framework over benchmark schemes, indicating that improved performance is
achievable without the need for optimizing MC and the radiation patterns of the
RIS on the fly, which can be rather cumbersome.
