  Exact solutions describing rotating black holes can provide significant
opportunities for testing modified theories of gravity, which are motivated by
the challenges posed by dark energy and dark matter. Starting with a spherical
Kiselev black hole as a seed metric, we construct rotating Kiselev black holes
within the $f(R,T)$ gravity framework using the revised Newman-Janis algorithm
- the $f(R,T)$ gravity-motivated rotating Kiselev black holes (FRKBH), which
encompasses, as exceptional cases, Kerr ($K=0$) and Kerr-Newman ($K=Q^2$) black
holes. These solutions give rise to distinct classes of black holes surrounded
by fluids while considering specific values of the equation-of-state parameter,
$w$, for viable choices for the $f(R,T)$ function. From the parameter space or
domain of existence of black holes defined by $a$ and $\gamma$ for FKRBH, we
discover that when $a_1<a<a_2$, there is a critical value $\gamma=\gamma_E$
which corresponds to extreme value black holes portrayed by degenerate
horizons. When $a<a_1$ ($a>a_2$), we encounter two distinct critical values
$\gamma=\gamma_{E1}, \; \gamma_{E2}$ with $\gamma_{E1}>\gamma_{E2}$ (or
$\gamma=\gamma_{E3},\; \gamma_{E4}$ with $\gamma_{E3}>\gamma_{E4}$. We delve
into the horizon and global structure of FKRBH spacetimes and examine their
dependence on parameters $w$ and $\gamma$. This exploration is motivated by the
remarkable effects of $f(R,T)$ gravity, which gives rise to diverse and
intricate spacetime structures within the domain where black holes exist.
