diff --git a/src/connection.c b/src/connection.c
index 564c976..54bbc07 100644
--- a/src/connection.c
+++ b/src/connection.c
@@ -86,12 +86,14 @@
 	if (!(conn->flags & CO_FL_POLL_SOCK))
 		__conn_sock_stop_both(conn);
 
-	/* The data layer might not be ready yet (eg: when using embryonic
-	 * sessions). If we're about to move data, we must initialize it first.
-	 * The function may fail and cause the connection to be destroyed, thus
-	 * we must not use it anymore and should immediately leave instead.
+	/* The connection owner might want to be notified about an end of
+	 * handshake indicating the connection is ready, before we proceed with
+	 * any data exchange. The callback may fail and cause the connection to
+	 * be destroyed, thus we must not use it anymore and should immediately
+	 * leave instead. The caller must immediately unregister itself once
+	 * called.
 	 */
-	if ((conn->flags & CO_FL_INIT_DATA) && conn->data->init(conn) < 0)
+	if (conn->xprt_done_cb && conn->xprt_done_cb(conn) < 0)
 		return;
 
 	if (conn->xprt && fd_send_ready(fd) &&
@@ -137,6 +139,16 @@
 	if (unlikely(!(conn->flags & (CO_FL_WAIT_L4_CONN | CO_FL_WAIT_L6_CONN | CO_FL_CONNECTED))))
 		conn->flags |= CO_FL_CONNECTED;
 
+	/* The connection owner might want to be notified about failures to
+	 * complete the handshake. The callback may fail and cause the
+	 * connection to be destroyed, thus we must not use it anymore and
+	 * should immediately leave instead. The caller must immediately
+	 * unregister itself once called.
+	 */
+	if (((conn->flags ^ flags) & CO_FL_NOTIFY_DONE) &&
+	    conn->xprt_done_cb && conn->xprt_done_cb(conn) < 0)
+		return;
+
 	/* The wake callback is normally used to notify the data layer about
 	 * data layer activity (successful send/recv), connection establishment,
 	 * shutdown and fatal errors. We need to consider the following
diff --git a/src/session.c b/src/session.c
index 62713ab..cc2a0b8 100644
--- a/src/session.c
+++ b/src/session.c
@@ -31,18 +31,8 @@
 struct pool_head *pool2_session;
 
 static int conn_complete_session(struct connection *conn);
-static int conn_update_session(struct connection *conn);
 static struct task *session_expire_embryonic(struct task *t);
 
-/* data layer callbacks for an embryonic stream */
-struct data_cb sess_conn_cb = {
-	.recv = NULL,
-	.send = NULL,
-	.wake = conn_update_session,
-	.init = conn_complete_session,
-	.name = "SESS",
-};
-
 /* Create a a new session and assign it to frontend <fe>, listener <li>,
  * origin <origin>, set the current date and clear the stick counters pointers.
  * Returns the session upon success or NULL. The session may be released using
@@ -251,11 +241,11 @@
 	 *          conn -- owner ---> task
 	 */
 	if (cli_conn->flags & CO_FL_HANDSHAKE) {
-		conn_attach(cli_conn, t, &sess_conn_cb);
+		conn_set_owner(cli_conn, t);
+		conn_set_xprt_done_cb(cli_conn, conn_complete_session);
 		t->process = session_expire_embryonic;
 		t->expire = tick_add_ifset(now_ms, p->timeout.client);
 		task_queue(t);
-		cli_conn->flags |= CO_FL_INIT_DATA;
 		return 1;
 	}
 
@@ -423,9 +413,14 @@
 	struct task *task = conn->owner;
 	struct session *sess = task->context;
 
+	conn_clear_xprt_done_cb(conn);
+
 	if (conn->flags & CO_FL_ERROR)
 		goto fail;
 
+	if (conn->flags & CO_FL_HANDSHAKE)
+		return 0; /* wait more */
+
 	/* if logs require transport layer information, note it on the connection */
 	if (sess->fe->to_log & LW_XPRT)
 		conn->flags |= CO_FL_XPRT_TRACKED;
@@ -439,8 +434,6 @@
 	if (!stream_new(sess, task, &conn->obj_type))
 		goto fail;
 
-	conn->flags &= ~CO_FL_INIT_DATA;
-
 	task_wakeup(task, TASK_WOKEN_INIT);
 	return 0;
 
@@ -449,21 +442,6 @@
 	return -1;
 }
 
-/* Update a session status. The connection is killed in case of
- * error, and <0 will be returned. Otherwise it does nothing.
- */
-static int conn_update_session(struct connection *conn)
-{
-	struct task *task = conn->owner;
-	struct session *sess = task->context;
-
-	if (conn->flags & CO_FL_ERROR) {
-		session_kill_embryonic(sess, task);
-		return -1;
-	}
-	return 0;
-}
-
 /*
  * Local variables:
  *  c-indent-level: 8