  We present a detailed study by Monte Carlo simulations and finite-size
scaling analysis of the phase diagram and ordered bulk phases for the
three-dimensional Blume-Capel antiferromagnet in the space of temperature and
magnetic and crystal fields (or two chemical potentials in an equivalent
lattice-gas model with two particle species and vacancies). The phase diagram
consists of surfaces of second- and first-order transitions that enclose a
``volume'' of ordered phases in the phase space. At relatively high
temperatures, these surfaces join smoothly along a line of tricritical points,
and at zero magnetic field we obtain good agreement with known values for
tricritical exponent ratios [Y. Deng and H.W.J. Bl{\"o}te, Phys. Rev. E {70},
0456111 (2004)]. In limited field regions at lower temperatures (symmetric
under reversal of the magnetic field), the tricritical line for this
three-dimensional model bifurcates into lines of critical endpoints and
critical points, connected by a surface of weak first-order transitions
{inside} the region of ordered phases. This phenomenon is not seen in the
two-dimensional version of the same model. We confirm the location of the
bifurcation as previously reported [Y.L. Wang and J.D. Kimel, J. Appl. Phys.
{69}, 6176 (1991)], and we identify the phases separated by this first-order
surface as antiferromagnetically (three-dimensional checker-board) ordered with
different vacancy densities. We visualize the phases by real-space snapshots
and by structure factors in the three-dimensional space of wave vectors.
