  We define a state space and a Markov process associated to the stochastic
quantisation equation of Yang-Mills-Higgs (YMH) theories. The state space
$\mathcal{S}$ is a nonlinear metric space of distributions, elements of which
can be used as initial conditions for the (deterministic and stochastic) YMH
flow with good continuity properties. Using gauge covariance of the
deterministic YMH flow, we extend gauge equivalence $\sim$ to $\mathcal{S}$ and
thus define a quotient space of "gauge orbits" $\mathfrak{O}$. We use the
theory of regularity structures to prove local in time solutions to the
renormalised stochastic YMH flow. Moreover, by leveraging symmetry arguments in
the small noise limit, we show that there is a unique choice of renormalisation
counterterms such that these solutions are gauge covariant in law. This allows
us to define a canonical Markov process on $\mathfrak{O}$ (up to a potential
finite time blow-up) associated to the stochastic YMH flow.
