  We study four-dimensional $N=4$ gauged supergravity with $SO(4)\times
SO(4)\sim SO(3)\times SO(3)\times SO(3)\times SO(3)$ gauge group in the
presence of symplectic deformations. There are in general four
electric-magnetic phases corresponding to each $SO(3)$ factor, but two phases
of the $SO(3)$ factors embedded in the $SO(6)$ R-symmetry are fixed. One phase
can be set to zero by $SL(2,\mathbb{R})$ transformations. The second one gives
equivalent theories for any non-vanishing values and can be set to
$\frac{\pi}{2}$ resulting in gauged supergravities that admit $N=4$
supersymmetric $AdS_4$ vacua. The remaining two phases are truely deformation
parameters leading to different $SO(4)\times SO(4)$ gauged supergravities. As
in the $\omega$-deformed $SO(8)$ maximal gauged supergravity, the cosmological
constant and scalar masses of the $AdS_4$ vacuum at the origin of the scalar
manifold with $SO(4)\times SO(4)$ symmetry do not depend on the
electric-magnetic phases. We find $N=1$ holographic RG flow solutions between
$N=4$ critical points with $SO(4)\times SO(4)$ and $SO(3)_{\textrm{diag}}\times
SO(3)\times SO(3)$ or $SO(3)\times SO(3)_{\textrm{diag}}\times SO(3)$
symmetries. We also give $N=2$ and $N=1$ RG flows from these critical points to
various non-conformal phases. However, contrary to the $\omega$-deformed
$SO(8)$ gauged supergravity, there exist non-trivial supersymmetric $AdS_4$
critical points only for particular values of the deformation parameters within
the scalar sectors under consideration.
