diff --git a/examples/eio/eio_connect_client.ml b/examples/eio/eio_connect_client.ml
index 5ade820..d0e2f21 100644
--- a/examples/eio/eio_connect_client.ml
+++ b/examples/eio/eio_connect_client.ml
@@ -25,21 +25,24 @@ let proxy_handler _env ~sw ~headers flow ~on_eof response _response_body =
        We'll be boring and use HTTP/1.1 again. *)
     let connection = Httpun_eio.Client.create_connection ~sw flow in
     let exit_cond = Eio.Condition.create () in
-    let response_handler =
-      http_handler ~on_eof:(fun () ->
-        Stdlib.Format.eprintf "http eof@.";
-        Eio.Condition.broadcast exit_cond;
-        on_eof ())
-    in
-    let request_body =
-      Httpun_eio.Client.request
-        ~flush_headers_immediately:true
-        ~error_handler:Httpun_examples.Client.error_handler
-        ~response_handler
-        connection
-        (Request.create ~headers `GET "/")
-    in
-    Body.Writer.close request_body
+    Eio.Fiber.fork ~sw (fun () ->
+      let response_handler =
+        http_handler ~on_eof:(fun () ->
+          Stdlib.Format.eprintf "http eof@.";
+          Eio.Condition.broadcast exit_cond;
+          on_eof ())
+      in
+      let request_body =
+        Httpun_eio.Client.request
+          ~flush_headers_immediately:true
+          ~error_handler:Httpun_examples.Client.error_handler
+          ~response_handler
+          connection
+          (Request.create ~headers `GET "/")
+      in
+      Body.Writer.close request_body);
+    Eio.Condition.await_no_mutex exit_cond;
+    Httpun_eio.Client.shutdown connection |> Eio.Promise.await
   | _response -> Stdlib.exit 124
 
 let main port proxy_host =
@@ -61,10 +64,15 @@ let main port proxy_host =
       let connection = Httpun_eio.Client.create_connection ~sw socket in
 
       let exit_cond = Eio.Condition.create () in
+      Eio.Fiber.fork ~sw (fun ()->
       let response_handler =
-        proxy_handler _env ~sw socket ~headers ~on_eof:(fun () ->
-          Stdlib.Format.eprintf "(connect) eof@.";
-          Eio.Condition.broadcast exit_cond)
+        fun response response_body ->
+          Eio.Fiber.fork ~sw @@ fun () ->
+          proxy_handler _env ~sw socket ~headers ~on_eof:(fun () ->
+            Stdlib.Format.eprintf "(connect) eof@.";
+            Eio.Condition.broadcast exit_cond)
+          response
+          response_body
       in
       let request_body =
         Httpun_eio.Client.request
@@ -77,7 +85,7 @@ let main port proxy_host =
       Body.Writer.close request_body;
       Eio.Condition.await_no_mutex exit_cond;
 
-      Httpun_eio.Client.shutdown connection |> Eio.Promise.await))
+      Httpun_eio.Client.shutdown connection |> Eio.Promise.await)))
 
 let () =
   let host = ref None in
diff --git a/lib/respd.ml b/lib/respd.ml
index 35c9c59..17389a2 100644
--- a/lib/respd.ml
+++ b/lib/respd.ml
@@ -104,12 +104,7 @@ let input_state t : Io_state.t =
     else if Body.Reader.is_read_scheduled response_body
     then Ready
     else Wait
-  | Upgraded _ ->
-    (* Upgraded is "Complete" because the descriptor doesn't wish to receive
-     * any more input.
-     * XXX(anmonteiro): not true for `CONNECT
-     *)
-    Wait
+  | Upgraded _ -> Wait
   | Closed -> Complete
 
 let output_state { request_body; state; writer; _ } : Io_state.t =
diff --git a/lib/response_state.ml b/lib/response_state.ml
index 3623991..80c3ba7 100644
--- a/lib/response_state.ml
+++ b/lib/response_state.ml
@@ -4,21 +4,23 @@ type t =
   | Streaming of Response.t * Body.Writer.t
   | Upgrade of Response.t * (unit -> unit)
 
-let output_state t ~request_method ~writer : Io_state.t =
-  match request_method with
-  | `CONNECT -> Wait
-  | _ ->
-    match t with
-    | Fixed _ -> Complete
+let output_state =
+  let response_sent_state = function
+    | `CONNECT -> Io_state.Wait
+    | _ -> Complete
+  in
+  fun t ~request_method ~writer : Io_state.t ->
+     match t with
+    | Upgrade _ -> Wait
     | Waiting ->
       if Serialize.Writer.is_closed writer then Complete
       else Wait
+    | Fixed _ -> response_sent_state request_method
     | Streaming(_, response_body) ->
-      if Serialize.Writer.is_closed writer then Complete
+      if Serialize.Writer.is_closed writer then response_sent_state request_method
       else if Body.Writer.requires_output response_body
       then Ready
-      else Complete
-    | Upgrade _ -> Wait
+      else response_sent_state request_method
 
 let flush_response_body t =
   match t with
diff --git a/lib_test/test_client_connection.ml b/lib_test/test_client_connection.ml
index 17e278c..f3e0a18 100644
--- a/lib_test/test_client_connection.ml
+++ b/lib_test/test_client_connection.ml
@@ -1866,6 +1866,41 @@ let test_read_response_before_shutdown () =
   connection_is_shutdown t;
 ;;
 
+let test_client_connect () =
+  let writer_woken_up = ref false in
+  let reader_woken_up = ref false in
+  let request' = Request.create
+    ~headers:(Headers.of_list ["host", "example.com:80"])
+    `CONNECT "/"
+  in
+  let t = create () in
+  let response = Response.create `OK in
+  let body =
+    request
+      t
+      request'
+      ~flush_headers_immediately:true
+      ~response_handler:(default_response_handler response)
+      ~error_handler:no_error_handler
+  in
+  write_request t request';
+  writer_yielded t;
+  Body.Writer.close body;
+  reader_ready t;
+  read_response t response;
+  reader_yielded t;
+  yield_reader t (fun () -> reader_woken_up := true);
+  writer_yielded t;
+  yield_writer t (fun () -> writer_woken_up := true);
+  Alcotest.(check bool) "Reader hasn't woken up yet" false !reader_woken_up;
+  Alcotest.(check bool) "Writer hasn't woken up yet" false !writer_woken_up;
+  shutdown t;
+  Alcotest.(check bool) "Reader woken up" true !reader_woken_up;
+  Alcotest.(check bool) "Writer woken up" true !writer_woken_up;
+  connection_is_shutdown t;
+;;
+
+
 let tests =
   [ "commit parse after every header line", `Quick, test_commit_parse_after_every_header
   ; "GET"         , `Quick, test_get
@@ -1914,4 +1949,5 @@ let tests =
   ; "shut down closes request body ", `Quick, test_read_response_before_shutdown
   ; "report exn during body read", `Quick, test_report_exn_during_body_read
   ; "read response after write eof", `Quick, test_can_read_response_after_write_eof
+  ; "Client support for CONNECT", `Quick, test_client_connect
   ]
diff --git a/lib_test/test_server_connection.ml b/lib_test/test_server_connection.ml
index d778749..22a0863 100644
--- a/lib_test/test_server_connection.ml
+++ b/lib_test/test_server_connection.ml
@@ -2438,6 +2438,24 @@ let test_write_response_after_read_eof () =
   connection_is_shutdown t;
 ;;
 
+let test_connect_method () =
+  let upgraded = ref false in
+  let upgrade_handler reqd =
+    Reqd.respond_with_upgrade reqd Headers.empty (fun () ->
+     upgraded := true)
+  in
+  let t = create ~error_handler upgrade_handler in
+  read_request
+    t
+    (Request.create
+      ~headers:(Headers.of_list [ "host", "example.com:80" ])
+      `CONNECT
+      "/");
+  write_response ~msg:"Upgrade response written" t (Response.create `Switching_protocols);
+  Alcotest.(check bool) "Callback was called" true !upgraded;
+  reader_yielded t;
+;;
+
 let tests =
   [ "initial reader state"  , `Quick, test_initial_reader_state
   ; "shutdown reader closed", `Quick, test_reader_is_closed_after_eof
@@ -2520,4 +2538,5 @@ let tests =
   ; "can read more requests after write eof", `Quick, test_can_read_more_requests_after_write_eof
   ; "can read more requests after write eof (before response sent)", `Quick, test_can_read_more_requests_after_write_eof_before_send_response
   ; "write response after reader EOF", `Quick,test_write_response_after_read_eof
+  ; "CONNECT method", `Quick, test_connect_method
   ]