Let $\mathrm{Aut}(X)$ denote the group of biholomorphic autmorphisms of the (noncompact) complex manifold $X$. If $X$ and $Y$ are two (noncompact) complex manifolds, then $\mathrm{Aut}(X)$ and $\mathrm{Aut}(Y)$ and $\mathrm{Aut}(X)\times \mathrm{Aut}(Y)$ are subgroups of $\mathrm{Aut}(X\times Y)$. My question is: Are there reasonable conditions, under which $\mathrm{Aut}(X)\times \mathrm{Aut}(Y)$ is a subgroup of finite index in $\mathrm{Aut}(X\times Y)$? Of course I am interested in the nontrivial cases, i.e. when $\mathrm{Aut}(X\times Y)$ itself is not a finite group. You may also assume that $X$ and $Y$ are algebraic manifolds.

Easiest case that I know is when $X$ and $Y$ are nonisogenous abelian varieties. If you want infinite automorphism group $\DeclareMathOperator{\Aut}{Aut} \Aut(X\times Y)$, you'll need at least one of them to have complex multiplication with endomorphism rings $\DeclareMathOperator{\End}{End} \End(X)$ or $\End(Y)$ having infinite unit group, but that's easy enough to arrange. More generally, won't it be true that if $\sigma\in\Aut(X\times Y)$ does not come from $\Aut(X)\times\Aut(Y)$, then you get a nonconstant map $X\to Y$ via $X\xrightarrow{i\times y_0} X\times Y\xrightarrow{\sigma}X\times Y\xrightarrow{p_2}Y$? So if there are no nonconstant maps from $X$ to $Y$, you'll have $\Aut(X)\times\Aut(Y)=\Aut(X\times Y)$. (I'm pretty sure that this is right if $X$ and $Y$ are projective, not entirely sure about the noncompact case.) 


If $X$ or $Y$ is $\mathbb{D}^n$ (the unit complex ball) or $\mathcal{T}^n$ (Teichmuller space), then $Aut(X\times Y)$ will be finite index in $Aut(X)\times Aut(Y)$. This follows by considering the Kobayashi (pseudo)metric. On a hyperbolic domain, the Kobayashi metric is a Finsler metric. On a product $X\times Y$, the Kobayashi metric is pointwise the maximum of the two metrics by a theorem of Royden. This metric is a nondegenerate Finsler metric on these examples. The product structure will then be seen pointwise in the Finsler norm, so any holomorphic biautomorphism will have to locally preserve the product structure (this follows from Royden's theorem for the Teichmuller metric case, which is the Kobayashi metric of Teichmuller space). I think the same will hold for $X$ or $Y$ products of these metrics. 

