product of quasi-coherent sheaves is not a quasi-coherent sheaf

Question

Reading the basics of $\mathcal{O}_{X}$ modules, where $\left(X,\mathcal{O}_{X}\right)$ is a fixed ringed space, i understood that arbitrary direct products of quasi-coherent $\mathcal{O}_{X}$ modules are quasi-coherent $\mathcal{O}_{X}$ modules. I also know by a result of Gabber that $\mathsf{Qcoh}\left(X\right)$ is a Grothendieck category for an arbitrary scheme $X$ and consequently by (iV) theorem 48 of section 2.3 it is a complete category. But, i was wondering: is the product of quasi-coherent sheaves coincide with the product in the category of $\mathcal{O}_{X}$ modules? Or there is a counter-example that arbitrary products of quasi-coherent sheaves are not quasi-coherent?