  We study critical and magnetic properties of a bilayer Ising system
consisting of two triangular planes A and B, with the antiferromagnetic (AF)
coupling $J_{\rm A}$ and the ferromagnetic (FM) one $J_{\rm B}$ for the
respective layers, which are coupled by the interlayer interaction $J_{\rm AB}$
by using Monte Carlo simulations. When $J_{\rm A}$ and $J_{\rm B}$ are of the
same order, the unfrustrated FM plane orders first at a high temperature
$T_{c1} \sim J_{\rm B}$. The spontaneous FM order then exerts influence on the
other frustrated AF plane as an effective magnetic field, which subsequently
induces a ferrimagnetic order in this plane at low temperatures below $T_{c2}$.
When short-range order is developed in the AF plane while the influence of the
FM plane is still small, there appears a preemptive
Berezinskii-Kosterlitz-Thouless-like pseudocritical crossover regime just above
the ferrimagnetic phase transition point, where the short-distance behavior up
to a rather large length scale exponentially diverging in $\propto \JA / T$ is
controlled by a line of Gaussian fixed points at $T = 0$. In the crossover
region, a continuous variation in the effective critical exponent $4/9 \lesssim
\eta^{\rm eff} \lesssim 1/2$ is observed. The phase diagram by changing the
ratio $J_{\rm A}/J_{\rm B}$ is also investigated.
