The aircraft collision avoidance problem can be formulated using a decision-theoretic planning framework where the optimal behavior requires balancing the competing objectives of avoiding collision and adhering to a flight plan. Due to noise in the sensor measurements and the stochasticity of intruder state trajectories, a natural representation of the problem is as a partially-observable Markov decision process (POMDP), where the underlying state of the system is Markovian and the observations depend probabilistically on the state. Many algorithms for finding approximate solutions to POMDPs exist in the literature, but they typically require discretization of the state and observation spaces. This paper investigates the introduction of a sample-based representation of state uncertainty to an existing algorithm called Real-Time Belief Space Search (RTBSS), which leverages branch-and-bound pruning to make searching the belief space for the optimal action more efficient. The resulting algorithm, called Monte Carlo Real-Time Belief Space Search (MC-RTBSS), is demonstrated on encounter scenarios in simulation using a beacon-based surveillance system and a probabilistic intruder model derived from recorded radar data.