The emergence of Integrated Sensing and Communication (ISAC) in future 6G networks comes with a variety of challenges to be solved. One of those is clutter removal, which should be applied to remove the influence of unwanted components, scattered by the environment, in the acquired sensing signal. While legacy radar systems already implement different clutter removal algorithms, ISAC requires techniques that are tailored to the envisioned use cases and the specific challenges that communications deployments bring along, like phase noise due to clock errors between transmitter and receiver. To that end, in this work we introduce Clutter Removal with Acquisitions Under Phase Noise (CRAP). We propose to vectorize the time-frequency channel acquired in a radio frame in a high-dimensional space. In an offline clutter acquisition step, singular value decomposition is used to determine the major clutter components. At runtime, the clutter is then estimated and removed by a subspace projection of the acquired radio frame onto the clutter components. Simulation results prove that CRAP offers benefits over prior art techniques robust to phase noise. In particular, our proposal does not suppress zero Doppler information, thereby enabling the detection of slow targets. Moreover, we show CRAP's real-time applicability in a millimeter-wave ISAC proof of concept, where a pedestrian is tracked in a cluttered lab environment.