  The first-order nature of the chiral phase transition in QCD-like theories
can play crucial roles to address a dark side of the Universe, where the
created out-of equilibrium is essential to serve as cosmological and
astrophysical probes such as gravitational wave productions, which have
extensively been explored. This interdisciplinary physics is built based on a
widely-accepted conjecture that the thermal chiral phase transition in QCD-like
theories with massless (light) three flavors is of first order. We find that
such a first order feature may not hold, when ordinary or dark quarks are
externally coupled to a weak enough background field of photon or dark photon
(which we collectively call a ``magnetic" field). We assume that a weak
``magnetic" background field could be originated from some ``magnetogenesis" in
the early Universe. We work on a Nambu-Jona-Lasinio model which can describe
the chiral phase transition in a wide class of QCD-like theories. We show that
in the case with massless (light) three flavors, the first-order feature goes
away when $2 f_\pi^2 \lesssim eB ( \ll (4 \pi f_\pi)^2)$, where $eB$ is the
``magnetic" field strength and $f_\pi$ the pion decay constant at the vacuum.
This disappearance is the generic consequence of the presence of the
``magnetically" induced scale anomaly and the ``magnetic" catalysis for the
chiral symmetry breaking, and would impact or constrain modeling dark QCD
coupled to an external ``magnetic" field.
