Context. Island effect (IE) aberrations are induced by differential pistons, tips, and tilts between neighboring pupil segments on ground-based telescopes, which severely limit the observations of circumstellar environments on the recently deployed exoplanet imagers (e.g., VLT/SPHERE, Gemini/GPI, Subaru/SCExAO) during the best observing conditions. Caused by air temperature gradients at the level of the telescope spiders, these aberrations were recently diagnosed with success on VLT/SPHERE, but so far no complete calibration has been performed to overcome this issue. Aims. We propose closed-loop focal plane wavefront control based on the asymmetric Fourier pupil wavefront sensor (APF-WFS) to calibrate these aberrations and improve the image quality of exoplanet high-contrast instruments in the presence of the IE. Methods. Assuming the archetypal four-quadrant aperture geometry in 8 m class telescopes, we describe these aberrations as a sum of the independent modes of piston, tip, and tilt that are distributed in each quadrant of the telescope pupil. We calibrate these modes with the APF-WFS before introducing our wavefront control for closed-loop operation. We perform numerical simulations and then experimental tests on a real system using Subaru/SCExAO to validate our control loop in the laboratory and on-sky. Results. Closed-loop operation with the APF-WFS enables the compensation for the IE in simulations and in the laboratory for the small aberration regime. Based on a calibration in the near infrared, we observe an improvement of the image quality in the visible range on the SCExAO/VAMPIRES module with a relative increase in the image Strehl ratio of 37%. Conclusions. Our first IE calibration paves the way for maximizing the science operations of the current exoplanet imagers. Such an approach and its results prove also very promising in light of the Extremely Large Telescopes (ELTs) and the presence of similar artifacts with their complex aperture geometry.