Fault diagnosis is a crucial area of research in the industry due to diverse operating conditions that exhibit non-Gaussian, multi-mode, and center-drift characteristics. Currently, data-driven approaches are the main focus in the field, but they pose challenges for continuous fault classification and parameter updates of fault classifiers, particularly in multiple operating modes and real-time settings. Therefore, a pressing issue is to achieve real-time multi-mode fault diagnosis for industrial systems. To address this problem, this paper proposes a novel approach that utilizes an evidence reasoning (ER) algorithm to fuse information and merge outputs from different base classifiers. These base classifiers are developed using a broad learning system (BLS) to improve good fault diagnosis performance. Moreover, in this approach, the pseudo-label learning method is employed to update model parameters in real-time. To demonstrate the effectiveness of the proposed approach, we perform experiments using the multi-mode Tennessee Eastman process dataset.