  For every free product decomposition $G = G_{1} \ast ...\ast G_{q} \ast
F_{r}$ of a group of finite Kurosh rank $G$, where $F_r$ is a finitely
generated free group, we can associate some (relative) outer space
$\mathcal{O}$. We study the asymmetry of the Lipschitz metric $d_R$ on the
(relative) Outer space $\mathcal{O}$. More specifically, we generalise the
construction of Algom-Kfir and Bestvina, introducing an (asymmetric) Finsler
norm $\|\cdot\|^{L}$ that induces $d_R$. Let's denote by $Out(G, \mathcal{O})$
the outer automorphisms of $G$ that preserve the set of conjugacy classes of
$G_i$'s. Then there is an $Out(G, \mathcal{O})$-invariant function $\Psi :
\mathcal{O} \rightarrow \mathbb{R}$ such that when $\| \cdot \|^{L}$ is
corrected by $d \Psi$, the resulting norm is quasisymmetric. As an application,
we prove that if we restrict $d_R$ to the $\epsilon$-thick part of the relative
Outer space for some $\epsilon >0$, is quasi-symmetric . Finally, we generalise
for IWIP automorphisms of a free product a theorem of Handel and Mosher, which
states that there is a uniform bound which depends only on the group, on the
ratio of the relative expansion factors of any IWIP $\phi \in Out(F_n)$ and its
inverse.
