TITLE: Is there always a toric isomorphism between isomorphic toric varieties?
QUESTION [25 upvotes]: Suppose two toric varieties are isomorphic as abstract varieties. Does it follow that there exists a toric isomorphism between them?

Edit: the comments below lead me to believe that I'm using the terms "toric variety" and "toric morphism" in a non-standard way, so let me clarify. Here are the definitions I have in mind.
Definition: A toric variety is a normal variety $X$ together with (1) a dense open subvariety $T\subseteq X$, and (2) a group structure on $T$ making it a torus, such that the action of $T$ on itself extends to an action on $X$. (Note: I do not include the data of an isomorphism between $T$ and $\mathbb G_m^{\dim X}$, but only require that such an isomorphism exists.)
Definition: If $(X,T_X)$ and $(Y,T_Y)$ are toric varieties (group structures on $T_X$ and $T_Y$ implicit), a toric morphism between them is a morphism $f:X\to Y$ which restricts to a group homomorphism $f|_{T_X}:T_X\to T_Y$.

REPLY [10 votes]: I don't know if you are still interested in a reference for this, but I recently found that this is proved in section 4 of "Lifting of morphisms to quotient presentations" by F. Berchtold. The proof uses the Cox construction and it seems to me that the argument is the one that David suggested above, but I am not done reading the paper. I hope this helps.