  We perform the first systematic study on how dynamical stellar tides and
general relativistic (GR) effects affect the dynamics and outcomes of
binary-single interactions. For this, we have constructed an N-body code that
includes tides in the affine approximation, where stars are modeled as
self-similar ellipsoidal polytropes, and GR corrections using the commonly-used
post-Newtonian formalism. Using this numerical formalism, we are able resolve
the leading effect from tides and GR across several orders of magnitude in both
stellar radius and initial target binary separation. We find that the main
effect from tides is the formation of two-body tidal captures that form during
the chaotic and resonant evolution of the triple system. The two stars
undergoing the capture spiral in and merge. The inclusion of tides can thus
lead to an increase on the stellar coalescence rate. We also develop an
analytical framework for calculating the cross section of tidal inspirals
between any pair of objects with similar mass. From our analytical and
numerical estimates we find that the rate of tidal inspirals relative to
collisions increases as the initial semi-major axis of the target binary
increases and the radius of the interacting tidal objects decreases. The
largest effect is therefore found for triple systems hosting white dwarfs and
neutron stars. In this case, we find the rate of highly eccentric white dwarf -
neutron star mergers to likely be dominated by tidal inspirals. While tidal
inspirals occur rarely, we note that they can give rise to a plethora of
thermonuclear transients such as Ca-rich transients.
