A large part of the theory of Hardy spaces on products of Euclidean spaces has been extended to the setting of products of stratified Lie groups. This includes characterisation of \(\mathsf {H}^1\) by square functions and by atomic decompositions, proof of the duality of \(\mathsf {H}^1\) with \(\mathsf {BMO}\) , and description of many interpolation spaces. Until now, however, two aspects of the classical theory have been conspicuously absent: the characterisation of \(\mathsf {H}^1\) by singular integrals (of Christ–Geller type) or by (vertical or nontangential) maximal functions. In this paper we fill in these gaps by developing new techniques on products of stratified groups, using the ideas in Chen et al. (Flag Hardy space theory on Heisenberg groups and applications. arXiv:2102.07371, 2021) on the Heisenberg group with flag structure.