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Directed polymers in random environments describe a perturbation of the simple random walk given by a random disorder (environment). The partition functions of this model have been thoroughly investigated in recent years, also motivated by their link with the solution of the Stochastic Heat Equation. While classical results focus on space-time independent disorder, we consider a Gaussian environment with (critical) spatial correlations decaying as $|x|^{-2}$ times a slowly varying function. We show that a phase transition, analogous to that in the space-time independent case, still occurs: in the high temperature regime the log-partition function satisfies a central limit theorem, while it vanishes in law in the low temperature regime. Remarkably, the inverse temperature needs to be tuned differently from the independent case, where the scaling constant $\hat{\beta}$ emerges from a nontrivial multi-scale dependence in the second moment computation — the core technical challenge of the work. Based on a joint work with Clément Cosco (Paris Dauphine) and Anna Donadini (Milano-Bicocca).