  Ferromagnetic Ising systems with competing interactions are considered in the
presence of a random field. We find that in three space dimensions the
ferromagnetic phase is disordered by a random field which is considerably
smaller than the typical interaction strength between the spins. This is the
result of a novel disordering mechanism triggered by an underlying spin-glass
phase. Calculations for the specific case of the long-range dipolar
LiHo_xY_{1-x}F_4 compound suggest that the above mechanism is responsible for
the peculiar dependence of the critical temperature on the strength of the
random field and the broadening of the susceptibility peaks as temperature is
decreased, as found in recent experiments by Silevitch et al. [Nature (London)
448, 567 (2007)]. Our results thus emphasize the need to go beyond the standard
Imry-Ma argument when studying general random-field systems.
