  We formulate and characterize a new constraint for Auxiliary Field Quantum
Monte Carlo (AFQMC) applicable for general fermionic systems, which allows for
the accumulation of phase in the random walk but disallows walkers with a
magnitude of phase greater than $\pi$ with respect to the trial wave function.
For short imaginary times, before walkers accumulate sizable phase values, this
approach is equivalent to exact free projection, allowing one to observe the
accumulation of bias associated with the constraint and thus estimate its
magnitude \textit{a priori}. We demonstrate the stability of this constraint
over arbitrary imaginary times and system sizes, highlighting the removal of
noise due to the fermionic sign problem. Benchmark total energies for a variety
of weakly and strongly correlated molecular systems reveal a distinct bias with
respect to standard phaseless AFQMC, with a comparative increase in accuracy
given a sufficient quality of trial wave function for the set of studied cases.
We then take this constraint, referred to as linecut (lc-) AFQMC, and
systematically release it (lcR-AFQMC), providing a route to obtain a smooth
bridge between constrained AFQMC and the exact free projection results.
