  Initial conditions for relativistic heavy-ion collisions may be far from
equilibrium (i.e. there are large initial contributions from the shear stress
tensor and bulk pressure) but it is expected that on very short time scales the
dynamics converge to a universal attractor that defines hydrodynamic behavior.
Thus far, studies of this nature have only considered an idealized situation at
LHC energies (high temperatures $T$ and vanishing baryon chemical potential
$\mu_B=0$) but, in this work, we investigate for the first time how
far-from-equilibrium effects may influence experimentally driven searches for
the Quantum Chromodynamic critical point at RHIC. We find that the path to the
critical point is heavily influenced by far from equilibrium initial conditions
where viscous effects lead to dramatically different $\left\{T,\mu_B\right\}$
trajectories through the QCD phase diagram. We compare hydrodynamic equations
of motion with shear and bulk coupled together at finite $\mu_B$ for both DNMR
and phenomenological Israel-Stewart equations of motion and discuss their
influence on potential attractors at finite $\mu_B$ and their corresponding
$\left\{T,\mu_B\right\}$ trajectories.
