For a uniform space \((X,\mu )\) , \(s_\mu X\) denotes the well-known Samuel compactification of X. We say that a realcompactification \(\alpha X\) of X is a uniform realcompactification whenever it is a topological subspace of \(s_\mu X\) , i.e., \(X\subset \alpha X\subset s_\mu X\) . In this paper we are mainly concerned with the following equation: \(\begin{aligned} \alpha (X\times Y)=\alpha X\times \alpha Y \end{aligned}\) for three specific uniform realcompactifications: the Samuel realcompactification \(H(U_\mu (X))\) , the \(G_\delta \) -realcompactification K(X), and the countable-modification realcompactification \(e_\mu X\) . For each of these, we provide necessary and sufficient conditions for the corresponding equality to hold. Moreover, we give a characterization of those uniform realcompactifications of \((X,\mu )\) that are, in addition, \(G_\delta \) -closed in \(s_\mu X\) .
