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Gibbsian structure in random point fields has been a classical tool for studying their spatial properties. However, exact Gibbs property is available only in a relatively limited class of models, and it does not adequately address many random fields with a strongly dependent spatial structure. In this work, we provide a general framework for approximate Gibbsian structure for strongly correlated random point fields. These include processes that exhibit strong spatial rigidity, in particular, a certain one-parameter family of analytic Gaussian zero point fields, namely the $α$-GAFs. Our framework entails conditions that may be verified via finite particle approximations to the process, a phenomenon that we call an approximate Gibbs property. We show that these enable one to compare the spatial conditional measures in the infinite volume limit with Gibbs-type densities supported on appropriate singular manifolds, a phenomenon we refer to as a generalized Gibbs property. We demonstrate the scope of our approach by showing that a generalized Gibbs property holds with a logarithmic pair potential for the $α$-GAFs for any value of $α$. This establishes the level of rigidity of the $α$-GAF zero process to be exactly $\lfloor \frac{1}α \rfloor$, settling in the affirmative an open question regarding the existence of point processes with any specified level of rigidity. For processes such as the zeros of $α$-GAFs, which involve complex, many-body interactions, our results imply that the local behaviour of the random points still exhibits 2D Coulomb-type repulsion in the short range. Our techniques can be leveraged to estimate the relative energies of configurations under local perturbations, with possible implications for dynamics and stochastic geometry on strongly correlated random point fields.