  Glauber models based on nucleon--nucleon interactions are commonly used to
characterize the initial state in high-energy nuclear collisions, and the
dependence of its properties on impact parameter or number of participating
nucleons. In this paper, an extension to the Glauber model is presented, which
accounts for an arbitrary number of effective sub-nucleon degrees of freedom,
or active constituents, in the nucleons. Properties of the initial state, such
as the number of constituent participants and collisions, as well as
eccentricity and triangularity, are calculated and systematically compared for
different assumptions of how to distribute the sub-nuclear degrees of freedom
and for various collision systems. It is demonstrated that at high collision
energy the number of produced particles scales with an average number of
sub-nucleon degrees of freedom of between $3$ and $5$. The source codes for the
constituent Monte Carlo Glauber extension, as well as for the calculation of
the overlap area and participant density in a standard Glauber model, are made
publicly available.
