Criegee intermediates (CIs) are formed during the ozonolysis of unsaturated hydrocarbons in the troposphere. The fate of CIs is of critical importance to tropospheric oxidation chemistry, particularly in the context of radical and secondary organic aerosol formation. Using the high-level ab initio G4(MP2) method, we investigate the 1,4 hydrogen shift reaction in CIs formed from ozonolysis of two common biogenic hydrocarbons: isoprene and α-pinene. We consider the uncatalysed reaction, as well as the reaction catalysed by a water molecule and by sulphuric acid. We show that sulphuric acid is a very effective catalyst, leading to a barrierless tautomerization relative to the free reactants and to very low reaction barrier heights relative to the reactant complexes. In particular, we obtain reaction barrier heights of Δ H298‡=24.5 (isoprene CI) and 8.4 (α-pinene CI) kJ mol-1 relative to the reactant complexes. Given the reaction of OH radicals with SO2 in the troposphere can ultimately yield sulphuric acid, these findings may have significant consequences for current atmospheric chemical models for regions of high sulphur concentrations.