This paper is dedicated to the analysis of backward stochastic differential equations (BSDEs) with jumps, subject to an additional global constraint involving all the com-ponents of the solution. We study the existence and uniqueness of a minimal solution for these so-called constrained BSDEs with jumps via a penalization procedure. This new type of BSDE offers a nice and practical unifying framework to the notions of constrained BSDEs presented in [22] and BSDEs with constrained jumps introduced in [17]. More remarkably, the solution of a multidimensional Brownian reflected BSDE studied in [16] and [14] can also be represented via a well chosen one-dimensional con-strained BSDE with jumps. This last result is very promising from a numerical point of view for the resolution of high dimensional optimal switching problems and more generally for systems of coupled variational inequalities.