We study the modifications in the structure of the Bose-Einstein condensates analogue spacetime due to the incorporation of Gaussian continuous quantum measurements and Markovian feedback. By constructing a suitable family of unravelings which reproduces the Wiseman-Milburn feedback master equation, we obtain the corrections to the density and the phase of the wave function defining the dynamical evolution of the analogue spacetime metric. We show that for each realization of the stochastic process, the phase fluctuation satisfies a generalized Klein-Gordon equation, where the metric is now stochastic. In the limit of small continuous measurement and feedback strengths, we recover a massless Klein-Gordon equation, as in the standard case.