Coherent synchronization of coupled fiber lasers has been successfully reported in recent experiments. While these results may lead to a dramatic advances in laser technology, the mechanism by which these lasers synchronize is not understood. We analyze a recently proposed iterated map model of fiber laser arrays to explore this phenomenon. In particular, we look at synchronous solutions to the map when the gain fields are constant. Determining the stability of these solutions is analytically tractable for a number of different coupling schemes. We find that in configurations with very high symmetry, coherent synchronized states are either unstable or their stability is not robust enough for practical purposes. In contrast, a lower symmetry configuration yields surprisingly robust coherence. This coherence persists beyond the pumping threshold for which the gain fields become time dependent.