This manuscript will discuss a numerical method where the six equations of two-phase flow, the solid heat conduction equations, and the two equations that describe neutron diffusion and precursor concentration are solved together in a tightly coupled, nonlinear fashion for a simplified model of a nuclear reactor core. This approach has two important advantages. The first advantage is a higher level of accuracy. Because the equations are solved together in a single nonlinear system, the solution is more accurate than the traditional "operator split" approach where the two-phase flow equations are solved first, the heat conduction is solved second and the neutron diffusion is solved third, limiting the temporal accuracy to $1^{st}$ order because the nonlinear coupling between the physics is handled explicitly. The second advantage of the method described in this manuscript is that the time step control in the fully implicit system can be based on the timescale of the solution rather than a stability-based time step restriction like the material Courant limit required of operator-split methods. In this work, a pilot code was used which employs this tightly coupled, fully implicit method to simulate a reactor core. Results are presented from a simulated control rod movement which show $2^{nd}$ order accuracy in time. Also described in this paper is a simulated rod ejection demonstrating how the fastest timescale of the problem can change between the state variables of neutronics, conduction and two-phase flow during the course of a transient.