  We consider bond percolation on high-dimensional product graphs
$G=\square_{i=1}^tG^{(i)}$, where $\square$ denotes the Cartesian product. We
call the $G^{(i)}$ the base graphs and the product graph $G$ the host graph.
Very recently, Lichev showed that, under a mild requirement on the
isoperimetric properties of the base graphs, the component structure of the
percolated graph $G_p$ undergoes a phase transition when $p$ is around
$\frac{1}{d}$, where $d$ is the average degree of the host graph.
  In the supercritical regime, we strengthen Lichev's result by showing that
the giant component is in fact unique, with all other components of order
$o(|G|)$, and determining the sharp asymptotic order of the giant. Furthermore,
we answer two questions posed by Lichev: firstly, we provide a construction
showing that the requirement of bounded-degree is necessary for the likely
emergence of a linear order component; secondly, we show that the isoperimetric
requirement on the base graphs can be, in fact, super-exponentially small in
the dimension. Finally, in the subcritical regime, we give an example showing
that in the case of irregular high-dimensional product graphs, there can be a
polynomially large component with high probability, very much unlike the
quantitative behaviour seen in the Erd\H{o}s-R\'enyi random graph and in the
percolated hypercube, and in fact in any regular high-dimensional product
graphs, as shown by the authors in a companion paper.
