Global sections functor commutes with tensor product for $\Gamma$-acyclic sheaves?

Question

Let $X$ be a topological space, and $A$ a ring. Let $F,G$ be sheaves of $A$-modules on $X$ which are acyclic for the global sections functor $\Gamma$. Is it true that $$\Gamma(F\otimes G) = \Gamma(F)\otimes_A\Gamma(G)?$$