  We study the effect of adding interlayer coupling to the square lattice,
$J_1$-$J_2$ Heisenberg model in high external magnetic field. In particular, we
consider a cubic lattice formed from stacked $J_1$-$J_2$ layers, with
interlayer exchange coupling $J_{\text{C}}$. For the 2-dimensional model
($J_{\text{C}}=0$) it has been shown that a spin-nematic phase appears close to
the saturation magnetic field for the parameter range $-0.4 \lesssim J_2/J_1$
and $J_2>0$. We determine the phase diagram for 3-dimensional model at high
magnetic field by representing spin flips out of the saturated state as bosons,
considering the dilute boson limit and using the Bethe-Salpeter equation to
determine the first instability of the saturated paramagnet. Close to the
highly frustrated point $J_2/J_1\sim0.5$, we find that the spin-nematic state
is stable even for $|J_{\text{C}}/J_1|\sim 1$. For larger values of $J_2/J_1$,
interlayer coupling favors a broad, phase-separated region. Further increase of
$|J_{\text{C}}|$ stabilizes a collinear antiferromagnet, which is selected via
the order-by-disorder mechanism.
