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Type Ia supernovae (SNe Ia) arise from the thermonuclear explosions of white dwarfs in multiple-star systems. A rare subclass of SNe Ia exhibit signatures of interaction with circumstellar material (CSM), allowing for direct constraints on companion material. While most known events show evidence for dense nearby CSM identified via peak-light spectroscopy (as SNe Ia-CSM), targeted late-time searches have revealed a handful of cases exhibiting delayed CSM interaction with detached shells. Here we present the first all-sky search for late CSM interaction in SNe Ia using a new image subtraction pipeline for mid-infrared data from the NEOWISE space telescope. Analyzing a sample of  ≈8500 SNe Ia, we report evidence for late-time mid-infrared brightening in five previously overlooked events spanning subtypes SNe Iax, SNe Ia-91T, and super-Chandra SNe Ia. Our systematic search doubles the known sample and suggests that ≳0.05% of SNe Ia exhibit mid-infrared signatures of delayed CSM interaction. The mid-infrared light curves ubiquitously indicate the presence of multiple (or extended) detached CSM shells located at ≳10¹⁶–10¹⁷cm, containing 10⁻⁶to 10⁻⁴M_⊙of dust, with some sources showing evidence for new dust formation, possibly within the cold, dense shell of the ejecta. We do not detect interaction signatures in spectroscopic and radio follow-up; however, the limits are largely consistent with previously confirmed events given the sensitivity and observation phase. Our results highlight that CSM interaction is more prevalent than previously estimated from optical and ultraviolet searches and that mid-infrared synoptic surveys provide a unique window into this phenomenon.