To report on a micro computed tomography (micro-CT) system capable of x-ray phase contrast imaging and of increasing spatial resolution at constant magnification. The micro-CT system implements the edge illumination (EI) method, which relies on two absorbing masks with periodically spaced transmitting apertures in the beam path; these split the beam into an array of beamlets and provide sensitivity to the beamlets' directionality, i.e. refraction. In EI, spatial resolution depends on the width of the beamlets rather than on the source/detector point spread function (PSF), meaning that resolution can be increased by decreasing the mask apertures, without changing the source/detector PSF or the magnification. We have designed a dedicated mask featuring multiple bands with differently sized apertures and used this to demonstrate that resolution is a tuneable parameter in our system, by showing that increasingly small apertures deliver increasingly detailed images. Phase contrast images of a bar pattern-based resolution phantom and a biological sample (a mouse embryo) were obtained at multiple resolutions. The new micro-CT system could find application in areas where phase contrast is already known to provide superior image quality, while the added tuneable resolution functionality could enable more sophisticated analyses in these applications, e.g. by scanning samples at multiple scales.