Local phase-space Berry curvature and Hall transport in textured twisted bilayer graphene

Slow twist-angle and heterostrain textures in twisted bilayer graphene provide a natural route to phase-space Berry geometry. We show that a purely geometric tetrad/shift sector does not generate mixed Berry curvature once the spin connection is treated consistently. By contrast, a projected textured mini-Dirac cone in twisted bilayer graphene acquires genuine mixed phase-space curvature. We analyze a controlled local Hall-bar limit with a one-dimensional texture and mirror $M_x$. In that limit the pseudogauge gradient renormalizes only the longitudinal conductivity along the textured direction, while a dc nonlinear Hall response appears only when an additional valley-odd tilt generates a Berry-curvature dipole. Both geometric responses peak at the same local filling, $μ_{\rm loc}=\sqrt{2}\,m$, providing a gate-tunable fingerprint of their common origin. All local-cone parameters and their texture susceptibilities are extracted from a heterostrained Bistritzer-MacDonald model at $θ=1.3^\circ$. The tilt invoked in the transport estimates corresponds to heterostrain of only $0.03\%-0.2\%$, below values routinely imaged in devices. The resulting theory provides a local analytic framework for textured moiré Dirac materials and cleanly separates geometric, texture-induced, and transport-level ingredients relevant to realistic TBG.