  We describe an approach for many-body calculations with a finite-temperature,
grand canonical ensemble formalism using auxiliary-field quantum Monte Carlo
(AFQMC) with a self-consistent constraint to control the sign problem. The
usual AFQMC formalism of Blankenbecler, Scalapino, and Sugar suffers from the
sign problem with most physical Hamiltonians, as is well known. Building on
earlier ideas to constrain the paths in auxiliary-field space [Phys. Rev. Lett.
\textbf{83}, 2777 (1999)] and incorporating recent developments in
zero-temperature, canonical-ensemble methods, we discuss how a self-consistent
constraint can be introduced in the finite-temperature,
grand-canonical-ensemble framework. This together with several other
algorithmic improvements discussed here leads to a more accurate, more
efficient, and numerically more stable approach for finite-temperature
calculations. We carry out a systematic benchmark study in the two-dimensional
repulsive Hubbard model at $1/8$ doping. Temperatures as low as $T=1/80$ (in
units of hopping) are reached. The finite-temperature method is exact at very
high temperatures, and approaches the result of the zero-temperature
constrained-path AFQMC as temperature is lowered. The benchmark shows that
systematically accurate results are obtained for thermodynamic properties.
