  We analyze a modified kinetic Ising model, so called $q$-neighbor Ising
model, with Metropolis dynamics, [Phys. Rev. E 92, 052105], on a duplex clique
and a partially duplex clique. In the $q$-neighbor Ising model each spin
interacts only with $q$ spins randomly chosen from its whole neighborhood. In
the case of a duplex clique the change of a spin is allowed only if both levels
simultaneously induce this change. Due to the mean-field like nature of the
model we are able to derive the analytic form of transition probabilities and
solve the corresponding master equation. The existence of the second level
changes dramatically the character of the phase transition. In the case of the
monoplex clique, the $q$-neighbor Ising model exhibits continuous phase
transition for $q=3$, discontinuous phase transition for $q \ge 4$ and for
$q=1$ and $q=2$ the phase transition is not observed. On the other hand, in the
case of the duplex clique continuous phase transitions are observed for all
values of $q$, even for $q=1$ and $q=2$. Subsequently we introduce a partially
duplex clique, parametrized by $r \in [0,1]$, which allows us to tune the
network from monoplex ($r=0$) to duplex ($r=1$). Such a generalized topology,
in which a fraction $r$ of all nodes appear on both levels, allows to obtain
the critical value of $r=r^*(q)$ at which a tricriticality (switch from
continuous to discontinuous phase transition) appears.
