Real representation Define a representation $\rho$ of a finite group $G$ over a $\mathbb{C}$-vector space to be real if the space admits a basis for which matrix $\rho(g)$ has real coefficients $\forall g \in G$.
I have to show that for ever $\rho$ it is true that $\rho \otimes \rho^*$ is always real ($\rho^*$ is the dual representation).
I think I've got an answer but it's pretty ugly so i would like to know if there is a clever solution to this question.
 A: Let $V$ be a complex vector space, say that a map $C:V \to V$ is a real structure if it is a conjugate-linear involution, i.e. if $C^2=\mathrm{id}_V$ and $C(v+\lambda w)=C(v)+\overline{\lambda} C(w)$ for all $v,w \in V, \lambda \in \Bbb C$. In that case, we say that $C$ is a $G$-equivariant real structure if $C$ is $G$-equivariant.

Lemma If $V$ is a complex representation of a group $G$, then there is a real representation (by which I mean a representation on a real vector space) $W$ of $G$ such that $\Bbb C \otimes_{\Bbb R} W \cong V$ (this is equivalent to being a real representation as defined in the question) iff $V$ admits a $G$-equivariant real structure.

Proof from a sophisticated viewpoint, this is an instance of Galois descent, but let's give an elementary argument. If $\Bbb C \otimes_{\Bbb R} W \cong V$, then $C(z \otimes w)=\overline{z} \otimes w$ defines a $G$-equivariant real structure.
Suppose that $C$ is a $G$-equivariant real structure on $V$, then $W:= \{v \in V \mid C(v)=v\}$ is a $G$-stable $\Bbb R$ subspace of $V$ and one gets a $\Bbb C$-linear $G$-equivariant map $\varphi:\Bbb C \otimes_{\Bbb R} W \to V, z\otimes w \to zw$.
To see that this is an isomorphism, note that $V=W\oplus iW$, as we can write $v=\frac{v+C(v)}{2}+i\frac{v-C(v)}{2i}$ with $\frac{v+C(v)}{2},\frac{v-C(v)}{2i} \in W$ and clearly $W \cap iW = \{0\}$. One also has $\Bbb C= \Bbb R \oplus i \Bbb R$, so $\Bbb C \otimes_{\Bbb R} W \cong W \oplus iW$ and the isomorphism is compatible with this. (Note:This can be extended to an equivalence of suitable categories)
To apply this to the problem, let $G$ be a finite group and let $\rho:G \to \mathrm{GL}(V)$ be a finite-dimensional complex representation, then choose a $G$-invariant scalar product $\langle -,- \rangle$ on $V$, this defines a $G$-equivariant conjugate-linear isomorphism $\psi:V \to V^*$ given by $v \mapsto \langle v,-\rangle$ (assuming the convention that the inner product is linear in the second argument)
Now consider the map $C:V \otimes_{\Bbb C} V^*\to V \otimes_{\Bbb C}V^*, v\otimes \xi \mapsto \psi^{-1}(\xi) \otimes \psi(v)$, $C^2= \mathrm{id}$ and $C$ is conjugate-linear and $G$-equivariant, so by the lemma, $\rho \otimes \rho^*$ is real.
