Superconductivity in layered copper-oxide compounds emerges when charge carriers are added to antiferromagnetically-ordered CuO2 layers. The carriers destroy the antiferromagnetic order, but strong spin fluctuations persist throughout the superconducting phase and are intimately linked to super-conductivity. Neutron scattering measurements of spin fluctuations in hole-doped copper oxides have revealed an unusual `hour-glass feature in the momentum-resolved magnetic spectrum, present in a wide range of superconducting and non-superconducting materials. There is no widely-accepted explanation for this feature. One possibility is that it derives from a pattern of alternating spin and charge stripes, an idea supported by measurements on stripe-ordered La1.875Ba0.125CuO4. However, many copper oxides without stripe order also exhibit an hour-glass spectrum$. Here we report the observation of an hour-glass magnetic spectrum in a hole-doped antiferromagnet from outside the family of superconducting copper oxides. Our system has stripe correlations and is an insulator, which means its magnetic dynamics can conclusively be ascribed to stripes. The results provide compelling evidence that the hour-glass spectrum in the copper-oxide superconductors arises from fluctuating stripes.
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