(This question is crossposted from MSE, since there the question did not recieve any attention whatsoever.)

I would like to know if there is a description (or at least some sufficient condition known) of (Noetherian) schemes $X$ such that the category $\mathrm{QCoh}_X$ does have exact direct products. Of course, I do not mean affine schemes since for affine schemes, direct products are exact (kind of) trivially.

Seems to me that viable candidates would be projective schemes, since direct products in the category of graded modules are exact, and the well-known functor $M \mapsto \tilde{M}$ is not far from being an equivalence (at least in my mind) of the category of graded modules over a graded ring and the category of quasi-coherent sheaves over the projective scheme. However, I was told this is not true, i.e. that the direct products are not exact even on simple projective schemes.

I would also like to know some example when this fails, i.e. when product of epimorphisms of quasi-coherent sheaves is not an epimorphism (especially if the scheme is projective).

I only stumbled upon the fact that direct products of quasi-coherent sheaves are not exact in general in the introduction to this paper by L. Positselski (first paragraph of the introduction, in fact). However, I cannot find any further information, so any reference would be greatly appreciated.

A reference to some text treating these question I would find most helpful.

Thanks in advance for any help.