An important problem in computational neuroscience is to understand how networks of spiking neurons can carry out various computations underlying behavior. Balanced spiking networks (BSNs) provide a powerful framework for implementing arbitrary linear dynamical systems in networks of integrate-and-fire neurons. However, the classic BSN model requires near-instantaneous transmission of spikes between neurons, which is biologically implausible. Introducing realistic synaptic delays leads to an pathological regime known as "ping-ponging", in which different populations spike maximally in alternating time bins, causing network output to overshoot the target solution. Here we document this phenomenon and provide a novel solution: we show that a network can have realistic synaptic delays while maintaining accuracy and stability if neurons are endowed with conditionally Poisson firing. Formally, we propose two alternate formulations of Poisson balanced spiking networks: (1) a "local" framework, which replaces the hard integrate-and-fire spiking rule within each neuron by a "soft" threshold function, such that firing probability grows as a smooth nonlinear function of membrane potential; and (2) a "population" framework, which reformulates the BSN objective function in terms of expected spike counts over the entire population. We show that both approaches offer improved robustness, allowing for accurate implementation of network dynamics with realistic synaptic delays between neurons. Both Poisson frameworks preserve the coding accuracy and robustness to neuron loss of the original model and, moreover, produce positive correlations between similarly tuned neurons, a feature of real neural populations that is not found in the deterministic BSN. This work unifies balanced spiking networks with Poisson generalized linear models and suggests several promising avenues for future research.