  We investigate the formulation of Vasiliev's four-dimensional higher-spin
gravity in operator form, without making reference to one specific ordering.
More precisely, we make use of the one-to-one mapping between operators and
symbols thereof for a family of ordering prescriptions that interpolate between
and go beyond Weyl and normal orderings. This correspondence allows us to
perturbatively integrate the Vasiliev system in operator form and in a variety
of gauges. Expanding the master fields in inhomogenous symplectic group
elements, and letting products be controlled only by the group, we specify a
family of factorized gauges in which we are able to integrate the system to all
orders, producing exact solutions, including but not restricted to ones
presented previously in the literature; and then connect, at first order, to a
family of rotated Vasiliev gauges in which the solutions can be represented in
terms of Fronsdal fields. The gauge function responsible for the latter
transformation is explicitly constructed at first order. The analysis of the
system in various orderings is facilitated by an analytic continuation of
Gaussian symbols, by means of which one can distinguish and connect the two
branches of the metaplectic double cover and give a rationale to the properties
of the inner Klein operators as Gaussian delta sequences defining analytic
delta densities. As an application of some of the techniques here developed, we
evaluate twistor space Wilson line observables on our exact solutions and show
their independence from auxiliary constructs up to the few first subleading
orders in perturbation theory.
