There are some contexts, for instance the time-harmonic version of some evolution problem, in which it seems natural to study variational inequalities in complex vector spaces. To fix ideas, let $V := H^1_0(\Omega; \mathbb{C})$ where $\Omega$ is a bounded domain of $\mathbb{R}^n$ and let $K \subset V$ be a convex set; a possible model problem is to find $u \in K$ such that
$$ \operatorname{Re}a(u,v-u) \ge \operatorname{Re} (f, v-u)_{L^2(\Omega; \mathbb{C})} \qquad \forall v \in K, \qquad (*) $$
with $a(u,v) = \int_{\Omega} \nabla u \cdot \nabla \overline{v}$.
At first sight this formulation appears to be consistent since for example the characterization of the orthogonal projection in the complex setting has this form, however the functional-analytic setting is strange because mappings like ${L^2(\Omega; \mathbb{C})} \ni u \mapsto \operatorname{Re} (f, u)_{L^2(\Omega; \mathbb{C})}$ are additive, real linear but not complex (anti)linear. The other option is of course to uncouple all the variables in real and imaginary parts, resulting in a system of variational inequalities governed by ugly bilinear forms acting on $H^1_0(\Omega)^2 \times H^1_0(\Omega)^2$. Is it true that the second formulation is the only sound one or can we find the correct way to deal with $(*)$ (and/or suitably modify it)?