  Consider a simple graph $G$. We call a labeling $w:E(G)\cup V(G)\rightarrow
\{1, 2, \dots, s\}$ (\textit{total vertex}) \textit{product-irregular}, if all
product degrees $pd_G(v)$ induced by this labeling are distinct, where
$pd_G(v)=w(v)\times\prod_{e\ni v}w(e)$. The strength of $w$ is $s$, the maximum
number used to label the members of $E(G)\cup V(G)$. The minimum value of $s$
that allows some irregular labeling is called \textit{the total vertex product
irregularity strength} and denoted $tvps(G)$. We provide some general bounds,
as well as exact values for chosen families of graphs. Keywords:
product-irregular labeling, total vertex product irregularity strength,
vertex-distinguishing labeling.
