Action-supervised fine-tuning of vision-language-action (VLA) policies fits demonstrations effectively but constrains only the directions that change predicted actions, leaving visual structure consistent across action-equivalent states free to collapse. We formalize this as residual visual collapse along local action fibers and propose FiberTune, a training-time objective that preserves teacher-structured visual residuals without adding inference-time overhead. FiberTune uses an online action probe to estimate action-predictive feature directions, filters them from intermediate visual-token representations, and aligns the resulting probe-filtered residuals to a frozen visual teacher while regularizing their effective rank. Under identical training conditions, FiberTune improves over task-loss-only fine-tuning in every one of six controlled simulation settings spanning two benchmarks and two architectures (pi_0.5 and OpenVLA-OFT), as well as on physical SO-101 pick-place; representative gains include +10.7 percentage points SR(5) on long-horizon CALVIN ABC-to-D and physical SO-101 task success rising from 72.7% to 78.1%. Residual diagnostics show that these gains coincide with increased probe-filtered residual teacher alignment and effective rank, consistent with the action-fiber motivation.