Add a test that reproduces the bug

src/test/regress/expected/remove_coordinator.out

@@ -5,10 +5,14 @@ SELECT master_remove_node('localhost', :master_port);
 
 (1 row)
 
--- restore coordinator for the rest of the tests
+-- to silence -potentially flaky- "could not establish connection after" warnings in below test
-SELECT citus_set_coordinator_host('localhost', :master_port);
+SET client_min_messages TO ERROR;
- citus_set_coordinator_host
+-- to fail fast if the hostname is not resolvable
----------------------------------------------------------------------
+SET citus.node_connection_timeout to '1s';
-
+BEGIN;
-(1 row)
+  SET application_name TO 'new_app_name';
-
+  -- that should fail because of bad hostname & port
+  SELECT citus_add_node('200.200.200.200', 1, 200);
+ERROR:  connection to the remote node postgres@200.200.200.200:1 failed
+SSL SYSCALL error: EOF detected
+connection to server was lost

src/test/regress/sql/remove_coordinator.sql

@@ -1,5 +1,40 @@
 -- removing coordinator from pg_dist_node should update pg_dist_colocation
 SELECT master_remove_node('localhost', :master_port);
 
+-- to silence -potentially flaky- "could not establish connection after" warnings in below test
+SET client_min_messages TO ERROR;
+
+-- to fail fast if the hostname is not resolvable
+SET citus.node_connection_timeout to '1s';
+
+BEGIN;
+  SET application_name TO 'new_app_name';
+
+  -- that should fail because of bad hostname & port
+  SELECT citus_add_node('200.200.200.200', 1, 200);
+
+  -- Since above command failed, now Postgres will need to revert the
+  -- application_name change made in this transaction and this will
+  -- happen within abort-transaction callback, so we won't be in a
+  -- transaction block while Postgres does that.
+  --
+  -- And when the application_name changes, Citus tries to re-assign
+  -- the global pid but it does so only for Citus internal backends,
+  -- and doing so for Citus internal backends doesn't require being
+  -- in a transaction block and is safe.
+  --
+  -- However, for the client external backends (like us here), Citus
+  -- doesn't re-assign the global pid because it's not safe to do so
+  -- outside of a transaction block. This is because, it would require
+  -- performing a catalog access to retrive the local node id when the
+  -- cached local node is invalidated like what just happened here
+  -- because of the failed citus_add_node() call made above.
+  --
+  -- So by failing here (rather than crashing), we ensure this behavior.
+ROLLBACK;
+
+RESET client_min_messages;
+RESET citus.node_connection_timeout;
+
 -- restore coordinator for the rest of the tests
 SELECT citus_set_coordinator_host('localhost', :master_port);