In tightly coupled multiphysics simulations, the conventional Monte Carlo assumption of piecewise-constant material properties is no longer valid, as temperature and density fields vary continuously in space. Current approaches to this problem each impose limitations on accuracy, generality, or computational efficiency. This work introduces the Lagrange Error-Adaptive Collision Sampling (LEACS) method, which addresses spatially varying material properties during Monte Carlo neutron transport by dynamically evaluating the coupled finite element solution along the particle’s flight path. Rather than converting the multiphysics solution into a lower-order representation, LEACS accesses native element shape functions directly, with sample positions chosen analytically via the Lagrange error bound to optimally minimize error in the reconstructed collision probability distribution. LEACS is implemented in OpenMC and demonstrated on pin cell and assembly test cases and coupled with MOOSE thermal solutions. Analytical error analysis confirms that Lagrange-adaptive spacing consistently outperforms equidistant spacing, with the greatest gains observed for long axial flight paths. Spatial tally results suggest that as few as three sample points per flight segment are sufficient to capture the radial and axial impact of temperature gradients on the (n,γ) reaction rate distribution. Computational cost analysis reveals slowdown factors vary depending on the model geometry, with finite element point location and inverse mapping as the dominant cost components and primary targets for future optimization.