Add realtime start instructions config override (#14270)

- add `realtime_start_instructions` config support
- thread it into realtime context updates, schema, docs, and tests

codex-rs/core/tests/suite/compact_remote.rs

@@ -161,6 +161,25 @@ fn assert_request_contains_realtime_start(request: &responses::ResponsesRequest)
     );
 }
 
+fn assert_request_contains_custom_realtime_start(
+    request: &responses::ResponsesRequest,
+    instructions: &str,
+) {
+    let body = request.body_json().to_string();
+    assert!(
+        body.contains("<realtime_conversation>"),
+        "expected request to preserve the realtime wrapper"
+    );
+    assert!(
+        body.contains(instructions),
+        "expected request to use custom realtime start instructions"
+    );
+    assert!(
+        !body.contains("Realtime conversation started."),
+        "expected request to replace the default realtime start instructions"
+    );
+}
+
 fn assert_request_contains_realtime_end(request: &responses::ResponsesRequest) {
     let body = request.body_json().to_string();
     assert!(
@@ -1518,6 +1537,53 @@ async fn snapshot_request_shape_remote_pre_turn_compaction_restates_realtime_sta
     Ok(())
 }
 
+#[tokio::test(flavor = "multi_thread", worker_threads = 2)]
+async fn remote_request_uses_custom_experimental_realtime_start_instructions() -> Result<()> {
+    skip_if_no_network!(Ok(()));
+
+    let server = wiremock::MockServer::start().await;
+    let realtime_server = start_remote_realtime_server().await;
+    let custom_instructions = "custom realtime start instructions";
+    let mut builder = remote_realtime_test_codex_builder(&realtime_server).with_config({
+        let custom_instructions = custom_instructions.to_string();
+        move |config| {
+            config.experimental_realtime_start_instructions = Some(custom_instructions);
+        }
+    });
+    let test = builder.build(&server).await?;
+
+    let responses_mock = responses::mount_sse_once(
+        &server,
+        responses::sse(vec![
+            responses::ev_assistant_message("m1", "REMOTE_FIRST_REPLY"),
+            responses::ev_completed("r1"),
+        ]),
+    )
+    .await;
+
+    start_realtime_conversation(test.codex.as_ref()).await?;
+
+    test.codex
+        .submit(Op::UserInput {
+            items: vec![UserInput::Text {
+                text: "USER_ONE".to_string(),
+                text_elements: Vec::new(),
+            }],
+            final_output_json_schema: None,
+        })
+        .await?;
+    wait_for_event(&test.codex, |ev| matches!(ev, EventMsg::TurnComplete(_))).await;
+
+    assert_request_contains_custom_realtime_start(
+        &responses_mock.single_request(),
+        custom_instructions,
+    );
+
+    close_realtime_conversation(test.codex.as_ref()).await?;
+    realtime_server.shutdown().await;
+    Ok(())
+}
+
 #[tokio::test(flavor = "multi_thread", worker_threads = 2)]
 async fn snapshot_request_shape_remote_pre_turn_compaction_restates_realtime_end() -> Result<()> {
     skip_if_no_network!(Ok(()));