Puzzling observations of both thermal and dielectric responses in multi-silicate glasses at low temperatures T to static magnetic fields B have been reported in the last decade and call for an extension of the standard two-level systems tunneling model. An explanation is proposed, capable of capturing at the same time the T -and B-dependence of the specific heat Cp and of the dielectric constant ǫ in these glasses. This theory points to the existence of anomalous multi-welled tunneling systems in the glasses -alongside the standard two-level systems -and indications are given for glasses which should achieve larger electric magnetocapacitive enhancements.PACS numbers: 61.43. Fs, 77.22.Ch, 65.60.+a The last decade has seen much renewed interest in the physics of cold non-metallic glasses, materials displaying some universal physical properties attributed to the lowenergy excitations characterising most amorphous solids. The two-level systems (2LS) tunneling model (TM) [1] has been rather successful in explaining a variety of interesting phenomena dealing with the thermal, dielectric and acoustic properties of structural glasses at temperatures T < 1 K. Limitations and failures of the 2LS TM (treating cooperative motion in terms of single particles, mostly) on the other hand, have been also discussed [2].These materials do not present, normally, any remarkable magnetic-field response phenomena other than possibly a weak contribution from trace paramagnetic Feimpurities. It came thus as a great surprise when measurements showed [3] that in some multi-component silicate glasses (but not in pure a-SiO 2 ) one is able to observe changes in the dielectric constant ǫ(T, B) and already in magnetic fields B as weak as a few Oe. A typical glass showing a strong response has some 100 ppm Fe 3+ and the composition Al 2 O 3 -BaO-SiO 2 (in short AlBaSi-O, in this paper). Measurements made on a thick sol-gel fabricated AlBaSi-O film showed changes in δǫ/ǫ = [ǫ(B) − ǫ(0)]/ǫ(0) of order 10 −4 and characterised by an enhancement peaking around 0.03 T for 10 mK < T < 200 mK then followed by a reduction of ǫ for B > 0.1 T. A further enhancement was also observed at much higher fields (B > 10 T).Another, cleaner, multi-component silicate glass (borosilicate BK7, with 6 ppm Fe 3+ ) and a dirtier one (borosilicate Duran, with 120 ppm Fe 3+ ) have shown similar -but weaker -magnetic anomalies, |δǫ/ǫ| ∼ 10 −5 , seemingly excluding the paramagnetic impurities as their source [4]. Yet another multi-silicate glass, aSiO 2+x C y H z was investigated [5], confirming the unusual findings in AlBaSi-O. A convincing explanation for the unusual electric magnetocapacitance behavior of these cold glasses has not yet been found.Recently, some consensus has been gained by the idea of a coupling of the standard 2LS to the magnetic field via nuclei in the glasses carrying an electric quadrupole moment as well as a magnetic dipole one [6]. The nuclear mechanism is supported by some features of the observed magnetic-field dependence of the polarization...