  It is well known that the density and anisotropy profile in the inner regions
of a stellar system with positive phase-space distribution function are not
fully independent. Here we study the interplay between density profile and
orbital anisotropy at large radii in physically admissible (consistent) stellar
systems. The analysis is carried out by using two-component (n-gamma,gamma_1)
spherical self-consistent galaxy models, in which one density distribution
follows a generalized gamma profile with external logarithmic slope n, and the
other a standard gamma_1 profile (with external slope 4). The necessary and
sufficient conditions for phase-space consistency are determined analytically,
also in presence of a dominant massive central black hole, and the analytical
phase-space distribution function of (n-gamma,1) models, and of n-gamma models
with a central black hole, is derived for gamma=0,1,2. It is found that the
density slope in the external regions of a stellar system can play an important
role in determining the amount of admissible anisotropy: in particular, for
fixed density slopes in the central regions, systems with a steeper external
density profile can support more radial anisotropy than externally flatter
models. This is quantified by an inequality formally identical to the ``cusp
slope-central anisotropy'' theorem (An & Evans 2006), relating at all radii
(and not just at the center) the density logarithmic slope and the anisotropy
indicator in all Osipkov-Merritt systems [Abridged].
