A cryogenic achromatic half-wave plate (HWP) for submillimetre astronomical polarimetry has been designed, manufactured, tested and deployed in the Balloon-borne Large-Aperture Submillimeter Telescope for Polarimetry (BLASTPol). The design is based on the five-slab Pancharatnam recipe and it works in the wavelength range 200-600 μm, making it the broadest-band HWP built to date at (sub)millimetre wavelengths. The frequency behaviour of the HWP has been fully characterized at room and cryogenic temperatures with incoherent radiation from a polarizing Fourier transform spectrometer. We develop a novel empirical model, complementary to the physical and analytical ones available in the literature, that allows us to recover the HWP Mueller matrix and phase shift as a function of frequency and extrapolated to 4 K. We show that most of the HWP non-idealities can be modelled by quantifying one wavelength-dependent parameter, the position of the HWP equivalent axes, which is then readily implemented in a map-making algorithm. We derive this parameter for a range of spectral signatures of input astronomical sources relevant to BLASTPol, and provide a benchmark example of how our method can yield improved accuracy on measurements of the polarization angle on the sky at submillimetre wavelengths.