diff --git a/peps/pep-0797.rst b/peps/pep-0797.rst
index 3bd131278ad..2b62db3cfe3 100644
--- a/peps/pep-0797.rst
+++ b/peps/pep-0797.rst
@@ -30,6 +30,18 @@ For example:
         interp.prepare_main(file=proxy)
         interp.exec("file.write('I didn't expect the Spanish Inquisition')")
 
+Terminology
+===========
+
+This PEP uses the term "share", "sharing", and "shareable" to refer to objects
+that are *natively* shareable between interpreters. This differs from :pep:`734`,
+which uses these terms to also describe an object that supports the :mod:`pickle`
+module.
+
+In addition to the new :class:`~concurrent.interpreters.SharedObjectProxy` type,
+the list of natively shareable objects can be found in :ref:`the documentation
+<interp-object-sharing>`.
+
 Motivation
 ==========
 
@@ -69,21 +81,56 @@ implementing other methods to share objects between interpreters.
 Specification
 =============
 
-.. class:: concurrent.interpreters.SharedObjectProxy
+
+.. function:: concurrent.interpreters.share(obj)
+
+    Ensure *obj* is natively shareable.
+
+    If *obj* is natively shareable, this function does not create a proxy and
+    simply returns *obj*. Otherwise, *obj* is wrapped in an instance of
+    :class:`~concurrent.interpreters.SharedObjectProxy` and returned.
+
+    If *obj* has a :meth:`~object.__share__` method, the default behavior of
+    this function is overridden; the object's ``__share__`` method will be
+    called to convert *obj* into a natively shareable version of itself, which
+    will be returned by this function. If the object returned by ``__share__``
+    is not natively shareable, this function raises an exception.
+
+    The behavior of this function is roughly equivalent to:
+
+    .. code-block:: python
+
+        def share(obj):
+            if _is_natively_shareable(obj):
+                return obj
+
+            if hasattr(obj, "__share__"):
+                shareable = obj.__share__()
+                if not _is_natively_shareable(shareable):
+                    raise TypeError(f"__share__() returned unshareable object: {shareable!r}")
+
+                return shareable
+
+            return SharedObjectProxy(obj)
+
+
+.. class:: concurrent.interpreters.SharedObjectProxy(obj)
 
     A proxy type that allows access to an object across multiple interpreters.
-    This cannot be constructed from Python; instead, use the
-    :func:`~concurrent.interpreters.share` function.
+    Instances of this object are natively shareable between subinterpreters.
 
+    Unlike :func:`~concurrent.interpreters.share`, *obj* will always be wrapped,
+    even if it is natively shareable already or already a ``SharedObjectProxy``
+    instance. The object's :meth:`~object.__share__` method is not invoked if
+    it is available. Thus, prefer using ``share`` where possible.
 
-.. function:: concurrent.interpreters.share(obj)
 
-    Wrap *obj* in a :class:`~concurrent.interpreters.SharedObjectProxy`,
-    allowing it to be used in other interpreter APIs as if it were natively
-    shareable.
+.. function:: object.__share__()
 
-    If *obj* is natively shareable, this function does not create a proxy and
-    simply returns *obj*.
+   Return a natively shareable version of the current object. This includes
+   shared object proxies, as they are also natively shareable. Objects composed
+   of shared object proxies are also allowed, such as a :class:`tuple` whose
+   elements are :class:`~concurrent.interpreters.SharedObjectProxy` instances.
 
 
 Interpreter Switching
@@ -108,6 +155,26 @@ accessed in subinterpreters through a proxy:
     interp.exec("foo()")
 
 
+Method Proxying
+---------------
+
+Methods on a shared object proxy will switch to their owning interpreter when
+accessed. In addition, any arguments passed to the method are implicitly called
+with :func:`~concurrent.interpreters.share` to ensure they are shareable (only
+types that are not natively shareable are wrapped in a proxy). The same happens
+to the return value of the method.
+
+For example, the ``__add__`` method on an object proxy is roughly equivalent
+to the following code:
+
+.. code-block:: python
+
+    def __add__(self, other):
+        with self.switch_interpreter():
+            result = self.value.__add__(share(other))
+            return share(result)
+
+
 Multithreaded Scaling
 ---------------------