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Doubly intractable models are encountered in a number of fields, e.g. social networks, ecology and epidemiology. Inference for such models requires the evaluation of a likelihood function, whose normalising factor depends on the model parameters and is assumed to be computationally intractable. The normalising constant of the posterior distribution and the additional normalising factor of the likelihood function result in a so-called doubly intractable posterior, for which it is difficult to directly apply Markov chain Monte Carlo methods. We propose a signed pseudo-marginal Metropolis-Hastings algorithm with an unbiased block-Poisson estimator to sample from the posterior distribution of doubly intractable models. As the estimator can be negative, the algorithm targets the absolute value of the estimated posterior and uses an importance sampling estimator to ensure simulation-consistent estimates of the posterior mean of a function of the parameters. The importance sampling estimator can perform poorly when its denominator is close to zero. We derive a finite-sample concentration inequality that ensures, with high probability, that this pathological case does not occur. Our estimator for doubly intractable problems has three advantages over existing estimators. First, the estimator is well-suited for efficient parallelisation and vectorisation. Second, its structure is ideal for correlated pseudo-marginal methods, which are well known to dramatically increase sampling efficiency. Third, the estimator enables the derivation of heuristic guidelines for tuning its hyperparameters under simplifying assumptions. We demonstrate the superior performance of our method in the standard benchmark example that models correlated spatial data using the Ising model, as well as the Kent distribution model for spherical data.