Joint matrix decomposition problems appear frequently in statistics and engineering, with notable applications in learning latent variable models and tensor factorization. Joint triangularization of nearly diagonalizable matrices can be performed via orthogonal matrices, and the related nonconvex optimization problem is more tractable than alternative approaches. We carry out a perturbation analysis of the noisy joint matrix triangularization problem, and we derive a bound on the distance between any feasible approximate triangularizer and its noise-free counterpart. The bound is an a posteriori one, in the sense that it is based on quantities that are observed (input matrices and functions thereof), and moreover it is oblivious to the algorithm used and/or the properties of the feasible solution (e.g., proximity to critical points) that are typical of existing bounds in the literature. To our knowledge, this is the first a posteriori bound for joint matrix decomposition. We discuss possible applications and we compare with analogous a priori bounds for the same problem.