Currently all subject sets of a client certificate are treated as UTF8 strings using the ASN1_STRING_to_UTF8 function (https://github.com/python/cpython/blob/8d999cbf4adea053be6dbb612b9844635c4dfb8e/Modules/_ssl.c#L912), however RFC 5280 states that the UniqueIdentifier is of type BIT STRING. Passing a BIT STRING into the ASN1_STRING_to_UTF8 function seems to return -1 and subsequently fail a SSL handshake.

Traceback (most recent call last):
  File "/usr/lib64/python3.6/asyncio/events.py", line 145, in _run
    self._callback(*self._args)
  File "/usr/lib64/python3.6/site-packages/tornado/platform/asyncio.py", line 138, in _handle_events
    handler_func(fileobj, events)
  File "/usr/lib64/python3.6/site-packages/tornado/iostream.py", line 702, in _handle_events
    self._handle_read()
  File "/usr/lib64/python3.6/site-packages/tornado/iostream.py", line 1471, in _handle_read
    self._do_ssl_handshake()
  File "/usr/lib64/python3.6/site-packages/tornado/iostream.py", line 1431, in _do_ssl_handshake
    if not self._verify_cert(self.socket.getpeercert()):
  File "/usr/lib64/python3.6/ssl.py", line 860, in getpeercert
    return self._sslobj.getpeercert(binary_form)
  File "/usr/lib64/python3.6/ssl.py", line 610, in getpeercert
    return self._sslobj.peer_certificate(binary_form)
ssl.SSLError: unknown error (_ssl.c:959)

(line 959 of _ssl.c corresponds to https://github.com/python/cpython/blob/8d999cbf4adea053be6dbb612b9844635c4dfb8e/Modules/_ssl.c#L914)

https://github.com/pyca/cryptography/issues/3542 might be of interest for further discussion.