diff --git a/src/underworld3/function/_function.pyx b/src/underworld3/function/_function.pyx
index 590dca34..0f7c2e1b 100644
--- a/src/underworld3/function/_function.pyx
+++ b/src/underworld3/function/_function.pyx
@@ -368,6 +368,16 @@ def global_evaluate_nd(   expr,
     Users should typically use :func:`underworld3.function.global_evaluate`
     which provides automatic unit handling and a cleaner interface.
 
+    Contract: this is a faithful *parallel* counterpart of :func:`evaluate` —
+    a query point is interpolated wherever in the mesh it lands (on any rank),
+    a point just outside the mesh is extrapolated from its true nearest cell,
+    and ``check_extrapolated`` returns an inside/outside flag per point. The
+    result is independent of the number of ranks (up to the rank-local
+    extrapolation residual near partition seams). Points that no rank can
+    locate in-cell are resolved by a best-claim reduction over ranks (see the
+    out-of-domain block below); set ``GE_LOCAL_FALLBACK=0`` to restore the
+    legacy behaviour where such points returned silently-wrong values.
+
     Note it is not efficient to call this function to evaluate an expression at
     a single coordinate. Instead the user should provide a numpy array of all
     coordinates requiring evaluation.
@@ -520,6 +530,103 @@ def global_evaluate_nd(   expr,
         return_value[index, :, :] = data_container.array[:, :, :]
         return_mask[index] = is_extrapolated.array[:]
 
+    # ------------------------------------------------------------------
+    # Out-of-domain extrapolation — keep the parallel result a faithful
+    # match for the serial ``evaluate()`` contract: interpolate a point
+    # wherever it lands across ranks, extrapolate a point just outside the
+    # mesh, and flag inside/outside.
+    #
+    # After the migrate round-trip, a query point that NO rank could locate
+    # in one of its cells returns flagged-extrapolated but valued from
+    # whichever rank the bare dm.migrate happened to strand it on — typically
+    # a geometrically far, WRONG cell (the classic symptom is an annulus
+    # boundary point reading a value from the opposite side of the domain).
+    # Serial ``evaluate()`` instead extrapolates from the TRUE nearest cell.
+    # Restore that contract with a "best-claim" reduction over the (small,
+    # boundary-layer) stranded set:
+    #
+    #   1. allgather the extrapolated points so every rank holds the SAME
+    #      global set;
+    #   2. each rank reports, per point, its nearest-local-cell distance and
+    #      its LOCAL rbf extrapolation of the field there;
+    #   3. Allreduce(MIN distance) + Allreduce(MIN rank) tie-break picks the
+    #      rank whose nearest cell is globally closest, and Allreduce(SUM of
+    #      the winner-only value/flag) scatters that rank's extrapolation back.
+    #
+    # A point some rank actually contains (distance ~ 0) naturally wins, so
+    # only genuinely-stranded points are corrected. Cost is O(boundary points)
+    # — no dense global tree, no exhaustive search.
+    #
+    # DEADLOCK SAFETY — read before editing. Every collective here (allgather,
+    # Allreduce) runs unconditionally on the IDENTICAL global set on every
+    # rank, so all ranks stay in lockstep (n_ext_total is itself a reduced
+    # value, so the `> 0` guard is taken identically everywhere). The per-rank
+    # value MUST come from the LOCAL rbf path (rbf=True): the FE interpolation
+    # path (petsc_interpolate / DMInterpolation) is itself collective and would
+    # desync here, because each rank classifies the same global set against its
+    # own domain (different interior-point counts) → hang. Never route the
+    # fallback value through FE interpolation.
+    #
+    # Serial is left untouched (the serial path above already extrapolates from
+    # the true nearest cell). Escape hatch: GE_LOCAL_FALLBACK=0 restores the
+    # legacy (silently-wrong out-of-domain) behaviour; default on.
+    # ------------------------------------------------------------------
+    import os
+    if uw.mpi.size > 1 and os.environ.get("GE_LOCAL_FALLBACK", "1") not in (
+            "0", "off", "false", "no", ""):
+        from mpi4py import MPI
+
+        comm = uw.mpi.comm
+        ext_idx = np.where(return_mask[:, 0, 0])[0]
+        ext_coords = np.ascontiguousarray(coords_array[ext_idx], dtype=np.double)
+
+        counts = np.array(comm.allgather(ext_coords.shape[0]), dtype=int)
+        n_ext_total = int(counts.sum())
+
+        if n_ext_total > 0:
+            parts = comm.allgather(ext_coords)
+            all_ext = np.concatenate(
+                [p for p in parts if p.size], axis=0).reshape(n_ext_total, -1)
+
+            # This rank's local rbf extrapolation of the global set. NON-collective
+            # value path — see DEADLOCK SAFETY above (must be rbf=True, never FE).
+            ext_vals, ext_flag = evaluate_nd(
+                expr, all_ext, rbf=True, evalf=False, verbose=False,
+                check_extrapolated=True,)
+            ext_vals = np.ascontiguousarray(
+                np.asarray(ext_vals, dtype=np.double).reshape((n_ext_total,) + expr_shape))
+            ext_flag = np.asarray(ext_flag).reshape(n_ext_total).astype(np.int32)
+
+            # Nearest-local-cell distance for every point (local kd-tree query).
+            mesh._build_kd_tree_index()
+            dist2, _ = mesh._centroid_index.query(all_ext, k=1, sqr_dists=True)
+            dist2 = np.ascontiguousarray(np.asarray(dist2, dtype=np.double).ravel())
+
+            # Globally-nearest cell per point, lowest rank as the tie-break.
+            min_dist2 = np.empty(n_ext_total, dtype=np.double)
+            comm.Allreduce([dist2, MPI.DOUBLE], [min_dist2, MPI.DOUBLE], op=MPI.MIN)
+            my_claim = np.where(dist2 <= min_dist2 * (1.0 + 1e-12) + 1e-300,
+                                comm.rank, comm.size).astype(np.int32)
+            win_rank = np.empty(n_ext_total, dtype=np.int32)
+            comm.Allreduce([my_claim, MPI.INT], [win_rank, MPI.INT], op=MPI.MIN)
+            i_win = (win_rank == comm.rank)
+
+            # Winner contributes value+flag, everyone else zero; SUM selects it.
+            contrib_val = np.ascontiguousarray(
+                np.where(i_win[:, None, None], ext_vals, 0.0))
+            best_val = np.empty_like(contrib_val)
+            comm.Allreduce([contrib_val, MPI.DOUBLE], [best_val, MPI.DOUBLE], op=MPI.SUM)
+            contrib_flag = np.where(i_win, ext_flag, 0).astype(np.int32)
+            best_flag = np.empty(n_ext_total, dtype=np.int32)
+            comm.Allreduce([contrib_flag, MPI.INT], [best_flag, MPI.INT], op=MPI.SUM)
+
+            # Scatter this rank's segment of the global set back to its points.
+            offset = int(counts[:comm.rank].sum())
+            seg = slice(offset, offset + ext_coords.shape[0])
+            if ext_idx.size:
+                return_value[ext_idx, :, :] = best_val[seg]
+                return_mask[ext_idx, 0, 0] = best_flag[seg].astype(bool)
+
     if not check_extrapolated:
         return return_value
     else: