  The observation of the electroweak production of a $W$ boson and a photon in
association with two jets, using $pp$ collision data at the Large Hadron
Collider at a centre of mass energy of $\sqrt{s}=13$~TeV, is reported. The data
were recorded by the ATLAS experiment from 2015 to 2018 and correspond to an
integrated luminosity of 140 fb$^{-1}$. This process is sensitive to the
quartic gauge boson couplings via the vector boson scattering mechanism and
provides a stringent test of the electroweak gauge symmetry breaking of the
Standard Model. Events are selected if they contain one electron or muon,
missing transverse momentum, at least one photon, and two jets. Multivariate
techniques are used to distinguish the electroweak $W\gamma jj$ process from
irreducible background processes. The observed significance of the electroweak
$W\gamma jj$ process is well above six standard deviations, compared to an
expected significance of 6.3 standard deviations. Fiducial and differential
cross sections are measured in a fiducial phase space close to the detector
acceptance, which are in reasonable agreement with leading order Standard Model
predictions from MadGraph5+Pythia8 and Sherpa. The results are used to
constrain new physics effects in the context of an effective field theory.
