In this paper, we show that the concept of regular unitary dilation of a pair of commuting contractions is intimately related to Markovian-like subspaces with respect to a pair of commuting unitary operators. Starting from this observation, a two-dimensional Markovian-like representation problem previously introduced by Attasi is studied geometrically. By using the geometric properties of the state spaces and some elementary properties of two-variable Hardy spaces, the spectral domain description of regular and coregular minimal representations are given.