  The evolution of the gravitational-wave phase in the signal produced by
inspiralling binaries of compact stars is modified by the nonzero deformability
of the two stars. Hence, the measurement of these corrections has the potential
of providing important information on the equation of state of nuclear matter.
Extensive work has been carried out over the last decade to quantify these
corrections, but it has so far been restricted to stars with zero intrinsic
magnetic fields. While the corrections introduced by the magnetic tension and
magnetic pressure are expected to be subdominant, it is nevertheless useful to
determine the precise conditions under which these corrections become
important. To address this question, we have carried out a second-order
perturbative analysis of the tidal deformability of magnetised compact stars
under a variety of magnetic-field strengths and equations of state describing
either neutron stars or quark stars. Overall, we find that magnetically induced
corrections to the tidal deformability will produce changes in the
gravitational-wave phase evolution that are unlikely to be detected for
realistic magnetic field i.e., $B\sim 10^{10} - 10^{12}\,{\rm G}$. At the same
time, if the magnetic field is unrealistically large, i.e., $B\sim
10^{16}\,{\rm G}$, these corrections would produce a sizeable contribution to
the phase evolution, especially for quark stars. In the latter case, the
induced phase differences would represent a unique tool to measure the
properties of the magnetic fields, providing information that is otherwise hard
to quantify.
