Autocatalysis and chemical amplification are properties of living systems that can lead to increased responsiveness and to self-replication. Here we describe a synthetic system in which a unique form of reagent compartmentalization gives rise to nonlinear kinetics that are subject to the precise size- and shape-selectivity of the host. The reactivity is reminiscent of autocatalytic behaviour, in which there is no direct contact between reagents and products, and our approach offers a general way to impose complex chemical behaviour onto synthetic systems.