bpo-39161: Document multi-phase init modules under Py_NewInterpreter()

Doc/c-api/init.rst

@@ -1230,15 +1230,31 @@ function. You can create and destroy them using the following functions:
       single: Py_FinalizeEx()
       single: Py_Initialize()
 
-   Extension modules are shared between (sub-)interpreters as follows: the first
-   time a particular extension is imported, it is initialized normally, and a
-   (shallow) copy of its module's dictionary is squirreled away.  When the same
-   extension is imported by another (sub-)interpreter, a new module is initialized
-   and filled with the contents of this copy; the extension's ``init`` function is
-   not called.  Note that this is different from what happens when an extension is
-   imported after the interpreter has been completely re-initialized by calling
-   :c:func:`Py_FinalizeEx` and :c:func:`Py_Initialize`; in that case, the extension's
-   ``initmodule`` function *is* called again.
+   Extension modules are shared between (sub-)interpreters as follows:
+
+   *  For modules using multi-phase initialization,
+      e.g. :c:func:`PyModule_FromDefAndSpec`, a separate module object is
+      created and initialized for each interpreter.
+      Only C-level static and global variables are shared between these
+      module objects.
+
+   *  For modules using single-phase initialization,
+      e.g. :c:func:`PyModule_Create`, the first time a particular extension
+      is imported, it is initialized normally, and a (shallow) copy of its
+      module's dictionary is squirreled away.
+      When the same extension is imported by another (sub-)interpreter, a new
+      module is initialized and filled with the contents of this copy; the
+      extension's ``init`` function is not called.
+      Objects in the module's dictionary thus end up shared across
+      (sub-)interpreters, which might cause unwanted behavior (see
+      `Bugs and caveats`_ below).
+
+      Note that this is different from what happens when an extension is
+      imported after the interpreter has been completely re-initialized by
+      calling :c:func:`Py_FinalizeEx` and :c:func:`Py_Initialize`; in that
+      case, the extension's ``initmodule`` function *is* called again.
+      As with multi-phase initialization, this means that only C-level static
+      and global variables are shared between these modules.
 
    .. index:: single: close() (in module os)
 
@@ -1264,14 +1280,16 @@ process, the insulation between them isn't perfect --- for example, using
 low-level file operations like  :func:`os.close` they can
 (accidentally or maliciously) affect each other's open files.  Because of the
 way extensions are shared between (sub-)interpreters, some extensions may not
-work properly; this is especially likely when the extension makes use of
-(static) global variables, or when the extension manipulates its module's
-dictionary after its initialization.  It is possible to insert objects created
-in one sub-interpreter into a namespace of another sub-interpreter; this should
-be done with great care to avoid sharing user-defined functions, methods,
-instances or classes between sub-interpreters, since import operations executed
-by such objects may affect the wrong (sub-)interpreter's dictionary of loaded
-modules.
+work properly; this is especially likely when using single-phase initialization
+or (static) global variables.
+It is possible to insert objects created in one sub-interpreter into
+a namespace of another (sub-)interpreter; this should be avoided if possible.
+
+Special care should be taken to avoid sharing user-defined functions,
+methods, instances or classes between sub-interpreters, since import
+operations executed by such objects may affect the wrong (sub-)interpreter's
+dictionary of loaded modules. It is equally important to avoid sharing
+objects from which the above are reachable.
 
 Also note that combining this functionality with :c:func:`PyGILState_\*` APIs
 is delicate, because these APIs assume a bijection between Python thread states