The internal reaction-induced fluctuations which occur in catalytic CO oxidation on a Pt field emitter tip have been studied using field electron microscopy (FEM) as a spatially resolving method. The structurally heterogeneous Pt tip consists of facets of different orientations with nanoscale dimensions. The FEM resolution of roughly 2 nm corresponds to a few hundred reacting adsorbed particles whose variations in the density are imaged as brightness fluctuations. In the bistable range of the reaction one finds fluctuation-induced transitions between the two stable branches of the reaction kinetics. The fluctuations exhibit a behaviour similar to that of an equilibrium phase transition, i.e. the amplitude diverges upon approaching the bifurcation point terminating the bistable range of the reaction. Simulations with a hybrid Monte Carlo/mean-field model reproduce the experimental observations. Fluctuations on different facets are typically uncorrelated but within a single facet a high degree of spatial coherence is found.