  We consider a one-dimensional mean-field-hydrodynamic model of a
two-component degenerate Fermi gas in an external trap, each component
representing a spin state of the same atom. We demonstrate that the
interconversion between them (linear coupling), imposed by a resonant
electromagnetic wave, transforms the immiscible binary gas into a miscible
state, if the coupling constant, $\kappa $, exceeds a critical value, $ \kappa
_{\mathrm{cr}}$. The effect is predicted in a variational approximation, and
confirmed by numerical solutions. Unlike the recently studied model of a binary
BEC with the linear coupling, the components in the immiscible phase of the
binary fermion mixture never fill two separated domains with a wall between
them, but rather form anti-locked ($\pi $ -phase-shifted) density waves.
Another difference from the bosonic mixture is spontaneous breaking of symmetry
between the two components in terms of numbers of atoms in them, $N_{1}$ and
$N_{2}$. The latter effect is characterized by the parameter $\nu \equiv
(N_{1}-N_{2})/(N_{1}+N_{2}) $ (only $N_{1}+N_{2}$ is a conserved quantity), the
onset of miscibility at $\kappa \geq \kappa_{\mathrm{cr}}$ meaning a transition
to $\nu \equiv 0$. At $\kappa <\kappa_{\mathrm{cr}}$, $\nu $ features damped
oscillations as a function of $\kappa $. We also briefly consider an asymmetric
model, with a chemical-potential difference between the two components.
