The Clifford algebra $\mathcal{C}\ell _{1,2d-1}$ is central and simple (L), and hence has a unique faithful, irreducible representation (over $\mathbb{R}$) (A). Denote this representation by $\gamma :\mathcal{C}\ell _{1,2d-1}\rightarrow \mathfrak{gl}(V)$, for some complex vector space $V$, let $e_0,\ldots ,e_{2d-1}$ be an orthonormal basis for $\mathcal{C}\ell _{1,2d-1}$, and define $\gamma ^\mu :=\gamma (e_\mu )$.
I have been told that we may always choose the $\gamma$ matrices so that $ (\gamma ^\mu )^*=\gamma _\mu=\begin{cases}(-\gamma ^0,\vec{\gamma}) & \text{for signature }(-,+,\cdots ,+) \\ (\gamma ^0,-\vec{\gamma}) & \text{for signature }(+,-,\cdots ,-)\end{cases}, $ where $^*$ denotes the Hermitian conjugate.
($\vec{\gamma}=(\gamma ^1,\ldots ,\gamma ^{2d-1})$ is the spatial part of $\gamma ^\mu$.)
How does one show that there is always a representation in the equivalence class of the given $\gamma$ that satisfies this property? (To be clear, I was only told this to be true in the case relevant for physics, i.e. $d=2$, but presumably it is true in general).