We propose a novel notion of fluid bisimulation equivalence that allows one to compare and reduce the behavior of labeled fluid stochastic Petri nets (LFSPNs) while preserving their discrete and continuous properties. The underlying stochastic model for the discrete part of the LFSPNs is a continuous time Markov chain (CTMC). The performance analysis of the continuous part of the LFSPNs is accomplished via the associated stochastic fluid models. For the fluid bisimulation on the discrete markings of two LFSPNs, we require it to be a (strong) Markovian bisimulation. On the continuous markings, for every pair of Markovian bisimilar discrete markings, the fluid flow rates of the continuous places in the first LFSPN should coincide with those of the corresponding continuous places in the second LFSPN. We prove that the resulting fluid bisimulation equivalence preserves fluid density and distribution, as well as discrete and continuous performance measures.