Let $E,F$ and $G$ be three complex $C^{\infty}$-vector bundles of rank $r,s$ and $rs$.

(I am using the notation from Kobayashi - Differential geometry of complex vector bundles, VI §2)

Assume we have an isomorphism $f: E\otimes F \stackrel{\sim}{\longrightarrow} G$.

If $E$ and $G$ are equipped with Hermitian structures, that is we have $(E,h)$ and $(G,h'')$ where $h$ and $h''$ are positive definite Hermitian bilinear forms on $E$ resp. $G$.

Can we always find a Hermitian structure $h'\in Herm^{+}(F)$, depending on $h$ and $h''$, such that $f$ gets an isometry? That is: do we get an isometry:

$\tilde{f}: (E\otimes F,h\otimes h') \stackrel{\sim}{\longrightarrow} (G,h'')$?

Otherwise stated: can we solve the equation $h\otimes x=h''$ for $x\in Herm^{+}(F)$?