  We consider a fractional nonlinear wave equations (fNLW) with a general
power-type nonlinearity, on the two-dimensional torus. Our main goal is to
construct invariant global-in-time Gibbs dynamics for a renormalized fNLW. We
first construct the Gibbs measure associated with this equation by using the
variational approach of Barashkov and Gubinelli. We then prove almost sure
local well-posedness with respect to Gibbsian initial data, by exploiting the
second order expansion. Finally, we extend solutions globally in time using
Bourgain's invariant measure argument.
