Let G be a group, and $\rho : G \to GL(n, \mathbb{C})$ be a representation of $G$. Then we also get the conjugate representation $\rho^* : G \to GL(n, \mathbb{C})$, where $\rho^*(g) = \overline{\rho(g)}$. This definition is naïve, but it will suffice here.
We say that $\rho$ is a real representation if there exists some $M \in GL(n,\mathbb{C})$ such that $M\rho(g)M^{-1} \in GL(n,\mathbb{R}) ~\forall~ g \in G$. If $\rho$ is not real, but still $\rho \cong \rho^*$, we say that $\rho$ is pseudoreal.
I believe it is a fact (though I don't have a reference) that the tensor product of two pseudoreal representations is real, but I've never quite understood why. Is there an elementary proof, and/or an easy way to understand this?
Simple examples: The fundamental representation $\mathbf{2}$ of $SU(2)$ is pseudoreal, but $\mathbf{2}\otimes\mathbf{2} = \mathbf{3}\oplus\mathbf{1}$ is real (the $\mathbf{3}$ is the fundamental rep. of $SO(3)$, lifted to $SU(2) \cong \text{Spin}(3)$). Similarly, if we consider $SU(2)\times SU(2)$, then $(\mathbf{2}, \mathbf{2}) = (\mathbf{2},\mathbf{1})\otimes(\mathbf{1},\mathbf{2})$ is real, being the lift to $SU(2)\times SU(2) \cong \text{Spin}(4)$ of the fundamental rep. of $SO(4)$.