diff --git a/pyzx/pauliwebs.py b/pyzx/pauliwebs.py
new file mode 100644
index 00000000..0a63bd2c
--- /dev/null
+++ b/pyzx/pauliwebs.py
@@ -0,0 +1,55 @@
+# PyZX - Python library for quantum circuit rewriting 
+#        and optimization using the ZX-calculus
+# Copyright (C) 2024 - Aleks Kissinger and John van de Wetering
+
+# Licensed under the Apache License, Version 2.0 (the "License");
+# you may not use this file except in compliance with the License.
+# You may obtain a copy of the License at
+
+#    http://www.apache.org/licenses/LICENSE-2.0
+
+# Unless required by applicable law or agreed to in writing, software
+# distributed under the License is distributed on an "AS IS" BASIS,
+# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+# See the License for the specific language governing permissions and
+# limitations under the License.
+
+from .circuit import Circuit
+from .utils import EdgeType, VertexType
+from .simplify import gadgetize, to_rg
+from .graph.base import BaseGraph
+
+
+def preprocess(g: BaseGraph):
+    gadgetize(g)
+    to_rg(g) # TODO: i/o should stay Z
+    in_circ = Circuit()
+    out_circ = Circuit()
+    for j,i in enumerate(g.inputs()):
+        e = g.incident_edges(i)[0]
+        v = g.neighbors(i)[0]
+        p = g.phase(v)
+        if g.edge_type(e) == EdgeType.HADAMARD:
+            in_circ.add_gate('H', j)
+            g.set_edge_type(e, EdgeType.SIMPLE)
+        if p != 0:
+            g.set_phase(v, 0)
+            in_circ.add_gate("ZPhase", j, phase=p)
+
+    for j,o in enumerate(g.outputs()):
+        r = g.get_row(o)
+        g.set_row(o, r + 1)
+        e = g.incident_edges(o)[0]
+        v = g.neighbors(o)[0]
+        p = g.phase(v)
+        if p != 0:
+            g.set_phase(v, 0)
+            out_circ.add_gate("ZPhase", j, phase=p)
+
+        if g.edge_type(e) == EdgeType.HADAMARD:
+            out_circ.add_gate('H', j)
+        g.remove_edge(e)
+        
+        v1 = g.add_vertex(VertexType.Z, qubit=g.qubit(o), row=r)
+        g.add_edge((v,v1), EdgeType.HADAMARD)
+        g.add_edge((v1,o), EdgeType.HADAMARD)
\ No newline at end of file
diff --git a/pyzx/simplify.py b/pyzx/simplify.py
index 582580ae..77f3c5ba 100644
--- a/pyzx/simplify.py
+++ b/pyzx/simplify.py
@@ -32,7 +32,7 @@
 from optparse import Option
 from typing import List, Callable, Optional, Union, Generic, Tuple, Dict, Iterator, cast
 
-from .utils import EdgeType, VertexType, toggle_edge, vertex_is_zx, toggle_vertex
+from .utils import EdgeType, VertexType, phase_is_clifford, toggle_edge, vertex_is_zx, toggle_vertex
 from .rules import *
 from .graph.base import BaseGraph, VT, ET
 from .graph.multigraph import Multigraph
@@ -361,6 +361,17 @@ def to_rg(g: BaseGraph[VT,ET], select:Optional[Callable[[VT],bool]]=None, change
                 for e in g.incident_edges(v):
                     g.set_edge_type(e, toggle_edge(g.edge_type(e)))
 
+def gadgetize(g: BaseGraph):
+    """Convert every non-Clifford phase to a phase gadget"""
+    for v in list(g.vertices()):
+        p = g.phase(v)
+        if not phase_is_clifford(p) and g.vertex_degree(v) > 1:
+            x = g.add_vertex(VertexType.Z, -1, g.row(v))
+            y = g.add_vertex(VertexType.Z, -2, g.row(v))
+            g.add_edge((x, y), EdgeType.HADAMARD)
+            g.add_edge((v, x), EdgeType.HADAMARD)
+            g.set_phase(y, p)
+            g.set_phase(v, 0)
 
 def tcount(g: Union[BaseGraph[VT,ET], Circuit]) -> int:
     """Returns the amount of nodes in g that have a non-Clifford phase."""
@@ -369,7 +380,7 @@ def tcount(g: Union[BaseGraph[VT,ET], Circuit]) -> int:
     count = 0
     phases = g.phases()
     for v in g.vertices():
-        if phases[v]!=0 and phases[v].denominator > 2:
+        if not phase_is_clifford(phases[v]):
             count += 1
     return count