diff --git a/assembly_mesh_plugin/plugin.py b/assembly_mesh_plugin/plugin.py
index df9d95d..02fabb0 100644
--- a/assembly_mesh_plugin/plugin.py
+++ b/assembly_mesh_plugin/plugin.py
@@ -45,10 +45,9 @@ def get_tagged_gmsh(self):
         # All the solids in the current part should be added to the mesh
         for s in obj.moved(loc).Solids():
             # Add the current solid to the mesh
-
             with tempfile.NamedTemporaryFile(suffix=".brep") as temp_file:
                 s.exportBrep(temp_file.name)
-                gmsh.model.occ.importShapes(temp_file.name)
+                ps = gmsh.model.occ.importShapes(temp_file.name)
 
             # TODO find a way to check if the OCC in gmsh is compatible with the
             # OCC in CadQuery. When pip installed they tend to be incompatible
@@ -60,6 +59,16 @@ def get_tagged_gmsh(self):
 
             gmsh.model.occ.synchronize()
 
+            # Technically, importShapes could import multiple entities/dimensions, so filter those
+            vol_ents = []
+            for p in ps:
+                if p[0] == 3:
+                    vol_ents.append(p[1])
+
+            # Set the physical name to be the part name in the assembly for all the solids
+            ps2 = gmsh.model.addPhysicalGroup(3, vol_ents)
+            gmsh.model.setPhysicalName(3, ps2, f"{name.split('/')[-1]}")
+
             # All the faces in the current part should be added to the mesh
             for face in s.Faces():
                 # Face name can be based on a tag, or just be a generic name
diff --git a/tests/test_meshes.py b/tests/test_meshes.py
new file mode 100644
index 0000000..162fe74
--- /dev/null
+++ b/tests/test_meshes.py
@@ -0,0 +1,44 @@
+import gmsh
+import assembly_mesh_plugin.plugin
+from tests.sample_assemblies import (
+    generate_nested_boxes,
+    generate_simple_nested_boxes,
+    generate_test_cross_section,
+    generate_assembly,
+)
+
+
+def test_simple_assembly():
+    """
+    Tests to make sure that the most basic assembly works correctly with tagging.
+    """
+
+    # Create the basic assembly
+    assy = generate_simple_nested_boxes()
+
+    # Create a mesh that has all the faces tagged as physical groups
+    assy.saveToGmsh(mesh_path="tagged_mesh.msh")
+
+    gmsh.initialize()
+
+    gmsh.open("tagged_mesh.msh")
+
+    # Check the solids for the correct tags
+    physical_groups = gmsh.model.getPhysicalGroups(3)
+    for group in physical_groups:
+        # Get the name for the current volume
+        cur_name = gmsh.model.getPhysicalName(3, group[1])
+
+        assert cur_name in ["shell", "insert"]
+
+    # Check the surfaces for the correct tags
+    physical_groups = gmsh.model.getPhysicalGroups(2)
+    for group in physical_groups:
+        # Get the name for this group
+        cur_name = gmsh.model.getPhysicalName(2, group[1])
+
+        # Skip any groups that are not tagged explicitly
+        if "_surface_" in cur_name:
+            continue
+
+        assert cur_name in ["shell_inner-right", "insert_outer-right", "in_contact"]