With the development of artificial intelligence, the ability to capture the background characteristics of hyperspectral imagery (HSI) has improved, showing promising performance in hyperspectral anomaly detection (HAD) tasks. However, existing methods proposed in recent years still suffer from certain limitations: (1) Constraints are lacking in the deep feature learning process in terms of the issue of the absence of prior background and anomaly information. (2) Hyperspectral anomaly detectors with traditional self-supervised deep learning methods fail to ensure prioritized reconstruction of the background. (3) The architecture of fully connected deep networks in hyperspectral anomaly detectors leads to low utilization of spatial information and the destruction of the original spatial relationship in hyperspectral imagery and disregards the spectral correlation between adjacent pixels. (4) Hypotheses or assumptions for background and anomaly distributions restrict the performance of many hyperspectral anomaly detectors because the distributions of background land covers are usually complex and not assumable in real-world hyperspectral imagery. In consideration of the above problems, in this paper, we propose a novel fully convolutional auto-encoder based on dual clustering and latent feature adversarial consistency (FCAE-DCAC) for HAD, which is carried out with self-supervised learning-based processing. Firstly, density-based spatial clustering of applications with a noise algorithm and connected component analysis are utilized for successive spectral and spatial clustering to obtain more precise prior background and anomaly information, which facilitates the separation between background and anomaly samples during the training of our method. Subsequently, a novel fully convolutional auto-encoder (FCAE) integrated with a spatial–spectral joint attention (SSJA) mechanism is proposed to enhance the utilization of spatial information and augment feature expression. In addition, a latent feature adversarial consistency network with the ability to learn actual background distribution in hyperspectral imagery is proposed to achieve pure background reconstruction. Finally, a triplet loss is introduced to enhance the separability between background and anomaly, and the reconstruction residual serves as the anomaly detection result. We evaluate the proposed method based on seven groups of real-world hyperspectral datasets, and the experimental results confirm the effectiveness and superior performance of the proposed method versus nine state-of-the-art methods.