We propose a new method for improving the stability of concentric tube robots. Prior work sought to improve stability through laser-cut patterns that reduced $EI/GJ$, the ratio of flexural rigidity to torsional rigidity, but this strategy has always entailed a steep trade-off in overall robot stiffness. Instead, we show that stability can be improved much more efficiently by allowing transverse anisotropy (direction-dependent flexural rigidity, $EI_x\ne EI_y$). We provide a generalized robot model with this property and give its associated stability criterion. In the two-tube case, this provides a new version of the well-known $EI/GJ$ criterion, now generalized to reveal the independent effects of $EI_x$ and $EI_y$. It shows that the most efficient strategy to improve stability while preserving stiffness is to reduce the flexural rigidity about the axis of precurvature while maintaining high off-axis flexural rigidity and torsional rigidity. We validate the approach with a balanced-stiffness pair of laser-machined Nitinol tubes, demonstrating model accuracy and significant stability improvement while requiring less stiffness reduction than prior methods.