  We study two possible cosmological consequences of a first-order phase
transition in the temperature range of $1$ GeV to $10^3$ TeV: the generation of
a stochastic gravitational wave background (SGWB) within the sensitivity of the
Laser Interferometer Space Antenna (LISA) and, simultaneously, primordial
magnetic fields that would evolve through the Universe's history and could be
compatible with the lower bound from $\gamma$-ray telescopes on intergalactic
magnetic fields (IGMF) at present time. We find that, if even a small fraction
of the kinetic energy in sound waves is converted into MHD turbulence, a first
order phase transition occurring at temperature between $1$ and $10^6$ GeV can
give rise to an observable SGWB signal in LISA and, at the same time, an IGMF
compatible with the lower bound from the $\gamma$-ray telescope MAGIC, for all
proposed evolutionary paths of the magnetic fields throughout the radiation
dominated era. For two values of the fraction of energy density converted into
turbulence, $\varepsilon_{\rm turb}=0.1$ and $1$, we provide the range of
first-order phase transition parameters (strength $\alpha$, duration
$\beta^{-1}$, bubbles wall speed $v_w$, and temperature $T_*$), together with
the corresponding range of magnetic field strength $B$ and correlation length
$\lambda$, that would lead to the SGWB and IGMF observable with LISA and MAGIC.
The resulting magnetic field strength at recombination can also correspond to
the one that has been proposed to induce baryon clumping, previously suggested
as a possible way to ease the Hubble tension. In the limiting case
$\varepsilon_{\rm turb} \ll 1$, the SGWB is only sourced by sound waves,
however, an IGMF is still generated. We find that values as small as
$\varepsilon_{\rm turb} \sim O(10^{-13})$ (helical) and $O (10^{-9})$
(non-helical) can provide IGMF compatible with MAGIC's lower bound.
