We study joint chance constraints where the distribution of the uncertain parameters is only known to belong to an ambiguity set characterized by the mean and support of the uncertainties and by an upper bound on their dispersion. This setting gives rise to pessimistic (optimistic) ambiguous chance constraints, which require the corresponding classical chance constraints to be satisfied for every (for at least one) distribution in the ambiguity set. We demonstrate that the pessimistic joint chance constraints are conic representable and thus computationally tractable if (i) the constraint coefficients of the decisions are deterministic, (ii) the support set of the uncertain parameters is a cone, and (iii) their dispersion function is positively homogeneous. We also show that tractability is lost as soon as either of the conditions (i), (ii) or (iii) is relaxed in the mildest possible way. We further prove that the optimistic joint chance constraints are tractable if and only if (i) holds. To showcase the power of our tractability results, we solve large-scale project management and image reconstruction models to global optimality.