  We prove that $A_R(G) \otimes_R A_R(H) \cong A_R(G \times H)$, if $G$ and $H$
are Hausdorff ample groupoids. As part of the proof, we give a new universal
property of Steinberg algebras. We then consider the isomorphism problem for
tensor products of Leavitt algebras, and show that no diagonal-preserving
isomorphism exists between $L_{2,R} \otimes L_{3,R}$ and $L_{2,R} \otimes
L_{2,R}$. Indeed, there are no unexpected diagonal-preserving isomorphisms
between tensor products of finitely many Leavitt algebras. We give an easy
proof that every $*$-isomorphism of Steinberg algebras over the integers
preserves the diagonal, and it follows that $L_{2,\mathbb{Z}} \otimes
L_{3,\mathbb{Z}} \not \cong L_{2,\mathbb{Z}} \otimes L_{2,\mathbb{Z}}$ (as
$*$-rings).
