Resistive switching offers a promising route to universal electronic memory, potentially replacing current technologies that are approaching their fundamental limits. In many cases switching originates from the reversible formation and dissolution of nanometre-scale conductive filaments, which constrain the motion of electrons, leading to the quantisation of device conductance into multiples of the fundamental unit of conductance, G0. Such quantum effects appear when the constriction diameter approaches the Fermi wavelength of the electron in the medium - typically several nanometres. Here we find that the conductance of silicon-rich silica (SiOx) resistive switches is quantised in half-integer multiples of G0. In contrast to other resistive switching systems this quantisation is intrinsic to SiOx, and is not due to drift of metallic ions. Half-integer quantisation is explained in terms of the filament structure and formation mechanism, which allows us to distinguish between systems that exhibit integer and half-integer quantisation.