  This study explores the behavior of compact stars within the framework of
$f(R,L_m,T)$ gravity, focusing on the functional form $f(R,L_m,T) = R + \alpha
TL_m$. The modified Tolman-Oppenheimer-Volkoff (TOV) equations are derived and
numerically solved for several values of the free parameter $\alpha$ by
considering both quark and hadronic matter -- described by realistic equations
of state (EoSs). Furthermore, the stellar structure equations are adapted for
two different choices of the matter Lagrangian density (namely, $L_m= p$ and
$L_m= -\rho$), laying the groundwork for our numerical analysis. As expected,
we recover the traditional TOV equations in General Relativity (GR) when
$\alpha \rightarrow 0$. Remarkably, we found that the two choices for $L_m$
have appreciably different effects on the mass-radius diagrams. Results
showcase the impact of $\alpha$ on compact star properties, while final remarks
summarize key findings and discuss implications, including compatibility with
observational data from NGC 6397's neutron star. Overall, this research
enhances comprehension of $f(R,L_m,T)$ gravity's effects on compact star
internal structures, offering insights for future investigations.
