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Monolithic 3D (M3D) integration provides massive vertical integration through the use of nanoscale inter-layer vias (ILVs). However, high integration density and aggressive scaling of the inter-layer dielectric make ILVs especially prone to defects. We present a low-cost built-in self-test (BIST) method that requires only two test patterns to detect opens, stuck-at faults, and bridging faults (shorts) in ILVs. We also propose an extended BIST architecture for fault detection, called Dual-BIST, to guarantee zero ILV fault masking due to single BIST faults and negligible ILV fault masking due to multiple BIST faults. We analyze the impact of coupling between adjacent ILVs arranged in a 1D array in block-level partitioned designs. Based on this analysis, we present a novel test architecture called Shared-BIST with the added functionality of localizing single and multiple faults, including coupling-induced faults. We introduce a systematic clustering-based method for designing and integrating a delay bank with the Shared-BIST architecture for testing small-delay defects in ILVs with minimal yield loss. Simulation results for four two-tier M3D benchmark designs highlight the effectiveness of the proposed BIST framework.