  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 Hardy spaces by square functions and by atomic
decompositions, proof of the duality of Hardy spaces with BMO, and description
of many interpolation spaces. Until now, however, two aspects of the classical
theory have been conspicuously absent: the characterisation of Hardy spaces 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 of Chen, Cowling,
Lee, Li and Ottazzi on the Heisenberg group with flag structure.
