  Nematic phases, where electrons in a solid spontaneously break rotational
symmetry while preserving the translational symmetry, exist in several families
of unconventional superconductors [1, 2]. Although superconductivity mediated
by nematic fluctuations is well established theoretically [3-7], it has yet to
be unambiguously identified experimentally [8, 9]. A major challenge is that
nematicity is often intertwined with other degrees of freedom, such as
magnetism and charge order. The FeSe$_{1-x}$S$_x$ family of iron based
superconductors provides a unique opportunity to explore this concept, as it
features an isolated nematic phase that can be suppressed by sulfur
substitution at a quantum critical point (QCP) near $x_c = 0.17$, where nematic
fluctuations are the largest [10-12]. Here, we performed scanning tunneling
spectroscopy measurements to visualize Boguliubov quasiparticle interference
patterns, from which we determined the momentum structure of the
superconducting gap near the Brillouin zone $\Gamma$ point of
FeSe$_{0.81}$S$_{0.19}$. The results reveal an anisotropic, near nodal gap with
minima that are $45^\circ$ rotated with respect to the Fe-Fe direction,
characteristic of a nematic pairing interaction, contrary to the usual
isotropic gaps due to spin mediated pairing in other tetragonal Fe-based
superconductors. The results are also in contrast with pristine FeSe, where the
pairing is mediated by spin fluctuations and the gap minima are aligned with
the Fe-Fe direction. Therefore, the measured gap structure demonstrates not
only a fundamental change of the pairing mechanism across the phase diagram of
FeSe$_{1-x}$S$_x$, but it also indicates the existence of superconductivity
mediated by nematic fluctuations in FeSe$_{0.81}$S$_{0.19}$.
