Tensor network techniques, known for their low-rank approximation ability of large tensors and matrices, have recently been successfully applied to simple discrete ordinates, multigroup, neutron transport eigenvalue problems. Specifically, the tensor train (TT) technique was shown to lead to massive compression of large transport matrix operators and large speedups because of the ability to perform linear algebra on the compressed matrix operators. However, those problems were simple and involved only one material throughout the problem domain. This simplification substantially reduced the rank of the matrix operators. In this work, we extend the TT neutron transport to realistic nuclear reactor problems. We investigate the impact of the material cross-section spatial variation on the TT matrix compression, ranks, ability to solve the TT eigenvalue problem, and TT solution times when possible. We also investigate the impact of the TT tolerance, the maximum allowed error incurred by the compression, on the criticality eigenvalue. From a positive standpoint, we find that for more realistic problems, the compression of matrix transport operators is reduced when compared to the simpler problems but remains substantial. From a negative standpoint, however, we find the TT decomposition substantially degrades the solution and eigenvalue quality, potentially making these methods unacceptable for reactor physics problems as of publication. In addition, we find that solution times increase when compared to the single-material problems because of the rank growth of transport matrix operators when performing matrix operations in the TT form. In the worst cases, the TT linear solvers fail to converge and exhibit oscillations. We theorize that the efficiency of the TT linear solvers does not generalize for all neutron transport eigenvalue problems. We also theorize that preconditioning is required for successful convergence of the TT system of linear equations. Given the substantial compression of TT transport operators and their impacts on solution quality, TT methods might be better suited to work where rapid and inexpensive design iterations are necessary to scope out problems of interest if efficient and low-cost TT linear solvers can be developed.