  The sources which reionized the Intergalactic Medium by redshift ~6 are still
unknown. A severe constraint on the ionization process is the low emissivity
required to maintain the ionization in the Lyman-{\alpha} forest.
Simulation-calibrated observations suggest a production rate of at most only a
few photons per baryon. In this work, we present a new solution to this
"photon-starvation" problem using a weak background of cosmic magnetic fields,
which may be present as a consequence of early-Universe physics and subsequent
magneto-hydrodynamical amplification. If present, such magnetic fields can
induce density perturbations which are dominant on scales comparable to those
probed by measurements of hydrogen-absorption lines at redshifts z~2-5. We show
that a sub-nanoGauss magnetic field, coherent on scale ~1 Mpc with an almost
scale-invariant spectrum, is sufficient to produce significant impact on the
effective optical depth, the appearance of the Lyman-{\alpha} forest on quasar
spectra, the pixel-flux statistics and the power spectrum of transmitted flux.
We also show that such magnetic-field signatures are effectively erased when
the metagalactic photoionization rate is increased, hence relaxing the
constraint on the cosmic photon budget available for reionization.
