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We present the first exact simulation method for multidimensional reflected Brownian motion (RBM). Exact simulation in this setting is challenging because of the presence of correlated local-time-like terms in the definition of RBM. We apply recently developed so-called ε-strong simulation techniques (also known as tolerance-enforced simulation) which allowus to provide a piecewise linear approximation to RBM with ε (deterministic) error in uniform norm. A novel conditional acceptance--rejection step is then used to eliminate the error. In particular, we condition on a suitably designed information structure so that a feasible proposal distribution can be applied. [ABSTRACT FROM AUTHOR]