Omnidirectional micro aerial vehicles (OMAVs) are more capable of doing environmentally interactive tasks due to their ability to exert full wrenches while maintaining stable poses. However, OMAVs often incorporate additional actuators and complex mechanical structures to achieve omnidirectionality. Obtaining precise mathematical models is difficult, and the mismatch between the model and the real physical system is not trivial. The large model-plant mismatch significantly degrades overall system performance if a non-adaptive model predictive controller (MPC) is used. This work presents the $\mathcal{L}_1$-MPC, an adaptive nonlinear model predictive controller for accurate 6-DOF trajectory tracking of an overactuated tiltrotor hexacopter in the presence of model uncertainties and external disturbances. The $\mathcal{L}_1$-MPC adopts a cascaded system architecture in which a nominal MPC is followed and augmented by an $\mathcal{L}_1$ adaptive controller. The proposed method is evaluated against the non-adaptive MPC, the EKF-MPC, and the PID method in both numerical and PX4 software-in-the-loop simulation with Gazebo. The $\mathcal{L}_1$-MPC reduces the tracking error by around 90% when compared to a non-adaptive MPC, and the $\mathcal{L}_1$-MPC has lower tracking errors, higher uncertainty estimation rates, and less tuning requirements over the EKF-MPC. We will make the implementations, including the hardware-verified PX4 firmware and Gazebo plugins, open-source at https://github.com/HITSZ-NRSL/omniHex.