We investigate the means by which cold gas can accrete on to Milky Way mass galaxies from a hot corona of gas, using a new smoothed particle hydrodynamics code, 'SPHS'. We find that the 'cold clumps' seen in many classic SPH simulations in the literature are not present in our SPHS simulations. Instead, cold gas condenses from the halo along filaments that form at the intersection of supernovae-driven bubbles from previous phases of star formation. This positive feedback feeds cold gas to the galactic disc directly, fuelling further star formation. The resulting galaxies in the SPH and SPHS simulations differ greatly in their morphology, gas phase diagrams and stellar content. We show that the classic SPH cold clumps owe to a numerical thermal instability caused by an inability for cold gas to mix in the hot halo. The improved treatment of mixing in SPHS suppresses this instability leading to a dramatically different physical outcome. In our highest resolution SPHS simulation, we find that the cold filaments break up into bound, physically motivated clumps that form stars. The filaments are overdense by a factor of 10-100 compared to the surrounding gas, suggesting that the fragmentation results from a physical non-linear instability driven by the overdensity. This 'fragmenting filament' mode of disc growth has important implications for galaxy formation, in particular the role of star formation in bringing cold gas into disc galaxies. © 2013 The Authors. Published by Oxford University Press on behalf of the Royal Astronomical Society.