We present an experimental and theoretical study of the chaotic ionization of quasi-one-dimensional potassium Rydberg wavepackets via a phase-space turnstile mechanism. Turnstiles form a general transport mechanism for numerous chaotic systems, and this study explicitly illuminates their relevance to atomic ionization. We create time-dependent Rydberg wavepackets, subject them to alternating applied electric-field "kicks", and measure the electron survival probability. Ionization depends not only on the initial electron energy, but also on the phase-space position of the electron with respect to the turnstile --- that part of the electron packet inside the turnstile ionizes after the applied ionization sequence, while that part outside the turnstile does not. The survival data thus encode information on the geometry and location of the turnstile, and are in good agreement with theoretical predictions.