We investigate the structural dependencies in the bank-firm credit market of Spain under a multilayer network perspective. In particular, the original bipartite network is decomposed into different layers representing different industrial sectors. We then study the correlations between layers based on normalized measures of overlaps of links and weights of banks between layers. To assess the statistical significance of such correlations, we compare the observed values with the expected ones obtained from random graph models specifying only global constraints, i.e. the total degree or the total strength in single layers, and from configuration models capturing the intrinsic heterogeneity in the local constraints like the observed degree sequence and/or strength sequence in single layers. We find that, first, the raw dependencies between layers of the observed network are highly heterogeneous. Second, when evaluated against the null models, on the one hand, the rescaled correlations after filtering out the effects of the global constraints typically display no significant difference to the observed correlations. In addition, in the binary version, almost all correlations are still present after subtracting the effects of the observed degree sequences in all layers. On the other hand, the observed correlations are partially explained by the local constraints maintained in the weighted configuration models. All in all, comparing the observed network with all referenced null models, we find that the multilayer credit network under scrutiny has a significant, non-random structure of correlations that cannot be explained by more primitive network properties alone. In the binary case, such a non-random structure is, for instance, typically observed in the pairs of layers that have high levels of overlaps and correlations. In contrast, in the weighted case, patterns are found in different pairs of layers that have various levels of overlaps and correlations.