diff --git a/src/sage/geometry/polyhedron/backend_number_field.py b/src/sage/geometry/polyhedron/backend_number_field.py
index 48629c4add3..85a64764cc5 100644
--- a/src/sage/geometry/polyhedron/backend_number_field.py
+++ b/src/sage/geometry/polyhedron/backend_number_field.py
@@ -25,5 +25,103 @@
 
 
 class Polyhedron_number_field(Polyhedron_field, Polyhedron_base_number_field):
+    r"""
+    Polyhedra whose data can be converted to number field elements
 
-    pass
+    All computations are done internally using a fixed real embedded number field,
+    which is determined automatically.
+
+    INPUT:
+
+    - ``Vrep`` -- a list ``[vertices, rays, lines]`` or ``None``.
+
+    - ``Hrep`` -- a list ``[ieqs, eqns]`` or ``None``.
+
+    EXAMPLES:
+
+    TODO --- examples where input is in SR. Can copy from backend_normaliz.
+
+
+    TESTS:
+
+    Tests from backend_field -- here the data are already either in a number field or in AA.
+
+        sage: p = Polyhedron(vertices=[(0,0),(AA(2).sqrt(),0),(0,AA(3).sqrt())],            # optional - sage.rings.number_field
+        ....:                rays=[(1,1)], lines=[], backend='number_field', base_ring=AA)
+        sage: TestSuite(p).run()                                                            # optional - sage.rings.number_field
+
+        sage: K.<sqrt3> = QuadraticField(3)                                                 # optional - sage.rings.number_field
+        sage: p = Polyhedron([(0,0), (1,0), (1/2, sqrt3/2)], backend='number_field')        # optional - sage.rings.number_field
+        sage: TestSuite(p).run()                                                            # optional - sage.rings.number_field
+
+    Check that :trac:`19013` is fixed::
+
+        sage: K.<phi> = NumberField(x^2-x-1, embedding=1.618)                               # optional - sage.rings.number_field
+        sage: P1 = Polyhedron([[0,1], [1,1], [1,-phi+1]], backend='number_field')           # optional - sage.rings.number_field
+        sage: P2 = Polyhedron(ieqs=[[-1,-phi,0]], backend='number_field')                   # optional - sage.rings.number_field
+        sage: P1.intersection(P2)                                                           # optional - sage.rings.number_field
+        The empty polyhedron in (Number Field in phi with defining polynomial x^2 - x - 1 with phi = 1.618033988749895?)^2
+
+    Check that :trac:`28654` is fixed::
+
+        sage: Polyhedron(lines=[[1]], backend='number_field')
+        A 1-dimensional polyhedron in QQ^1 defined as the convex hull of 1 vertex and 1 line
+    """
+
+    def _init_from_Vrepresentation(self, vertices, rays, lines,
+                                   minimize=True, verbose=False):
+        """
+        Construct polyhedron from V-representation data.
+
+        INPUT:
+
+        - ``vertices`` -- list of points. Each point can be specified
+           as any iterable container of
+           :meth:`~sage.geometry.polyhedron.base.base_ring` elements.
+
+        - ``rays`` -- list of rays. Each ray can be specified as any
+          iterable container of
+          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.
+
+        - ``lines`` -- list of lines. Each line can be specified as
+          any iterable container of
+          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.
+
+        - ``verbose`` -- boolean (default: ``False``). Whether to print
+          verbose output for debugging purposes.
+
+        EXAMPLES::
+
+            sage: p = Polyhedron(ambient_dim=2, backend='number_field')
+            sage: from sage.geometry.polyhedron.backend_number_field import Polyhedron_number_field
+            sage: Polyhedron_number_field._init_from_Vrepresentation(p, [(0,0)], [], [])
+        """
+        # TODO: Transform data
+        super()._init_from_Vrepresentation(vertices, rays, lines,
+                                           minimize=minimize, verbose=verbose)
+
+    def _init_from_Hrepresentation(self, ieqs, eqns, minimize=True, verbose=False):
+        """
+        Construct polyhedron from H-representation data.
+
+        INPUT:
+
+        - ``ieqs`` -- list of inequalities. Each line can be specified
+          as any iterable container of
+          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.
+
+        - ``eqns`` -- list of equalities. Each line can be specified
+          as any iterable container of
+          :meth:`~sage.geometry.polyhedron.base.base_ring` elements.
+
+        - ``verbose`` -- boolean (default: ``False``). Whether to print
+          verbose output for debugging purposes.
+
+        TESTS::
+
+            sage: p = Polyhedron(ambient_dim=2, backend='number_field')
+            sage: from sage.geometry.polyhedron.backend_number_field import Polyhedron_number_field
+            sage: Polyhedron_number_field._init_from_Hrepresentation(p, [(1, 2, 3)], [])
+        """
+        # TODO: Transform data
+        super()._init_from_Hrepresentation(ieqs, eqns, minimize=minimize, verbose=verbose)
diff --git a/src/sage/geometry/polyhedron/face.py b/src/sage/geometry/polyhedron/face.py
index d9dd5a6c4d6..421228813a9 100644
--- a/src/sage/geometry/polyhedron/face.py
+++ b/src/sage/geometry/polyhedron/face.py
@@ -1066,7 +1066,7 @@ def combinatorial_face_to_polyhedral_face(polyhedron, combinatorial_face):
         # Equations before inequalities in Hrep.
         H_indices = tuple(range(n_equations))
         H_indices += tuple(x+n_equations for x in combinatorial_face.ambient_H_indices(add_equations=False))
-    elif polyhedron.backend() in ('normaliz', 'cdd', 'field', 'polymake'):
+    elif polyhedron.backend() in ('normaliz', 'cdd', 'field', 'number_field', 'polymake'):
         # Equations after the inequalities in Hrep.
         n_ieqs = polyhedron.n_inequalities()
         H_indices = tuple(x for x in combinatorial_face.ambient_H_indices(add_equations=False))