diff --git a/src/backend/commands/repack.c b/src/backend/commands/repack.c
index bafdca80810..ce5b99c38b1 100644
--- a/src/backend/commands/repack.c
+++ b/src/backend/commands/repack.c
@@ -62,6 +62,7 @@
 #include "miscadmin.h"
 #include "optimizer/optimizer.h"
 #include "pgstat.h"
+#include "replication/logicalrelation.h"
 #include "storage/bufmgr.h"
 #include "storage/lmgr.h"
 #include "storage/predicate.h"
@@ -919,14 +920,12 @@ check_concurrent_repack_requirements(Relation rel, Oid *ident_idx_p)
 
 	/*
 	 * Obtain the replica identity index -- either one that has been set
-	 * explicitly, or the primary key.  If none of these cases apply, the
-	 * table cannot be repacked concurrently.  It might be possible to have
-	 * repack work with a FULL replica identity; however that requires more
-	 * work and is not implemented yet.
+	 * explicitly, or a non-deferrable primary key.  If none of these cases
+	 * apply, the table cannot be repacked concurrently.  It might be possible
+	 * to have repack work with a FULL replica identity; however that requires
+	 * more work and is not implemented yet.
 	 */
-	ident_idx = RelationGetReplicaIndex(rel);
-	if (!OidIsValid(ident_idx) && OidIsValid(rel->rd_pkindex))
-		ident_idx = rel->rd_pkindex;
+	ident_idx = GetRelationIdentityOrPK(rel);
 	if (!OidIsValid(ident_idx))
 		ereport(ERROR,
 				errcode(ERRCODE_OBJECT_NOT_IN_PREREQUISITE_STATE),
diff --git a/src/test/regress/expected/cluster.out b/src/test/regress/expected/cluster.out
index 96089bb0fa2..8f80f9f752d 100644
--- a/src/test/regress/expected/cluster.out
+++ b/src/test/regress/expected/cluster.out
@@ -796,15 +796,63 @@ ORDER BY o.relname;
  clstr_3
 (2 rows)
 
--- concurrently disallowed in catalogs
+--
+-- Check concurrent mode requirements
+--
+-- Disallowed in catalogs
 REPACK (CONCURRENTLY) pg_class;
 ERROR:  cannot repack relation "pg_class"
 HINT:  REPACK CONCURRENTLY is not supported for catalog relations.
--- CONCURRENTLY doesn't like partitioned tables
+-- Doesn't like partitioned tables
 REPACK (CONCURRENTLY) clstrpart;
 ERROR:  REPACK (CONCURRENTLY) is not supported for partitioned tables
 HINT:  Consider running the command on individual partitions.
+-- Doesn't support catalog tables
+REPACK (CONCURRENTLY) pg_class;
+ERROR:  cannot repack relation "pg_class"
+HINT:  REPACK CONCURRENTLY is not supported for catalog relations.
+-- Only support permanent tables, temp and unlogged tables are not supported
+CREATE TEMP TABLE repack_conc_temp (i int PRIMARY KEY);
+REPACK (CONCURRENTLY) repack_conc_temp;
+ERROR:  cannot repack relation "repack_conc_temp"
+HINT:  REPACK CONCURRENTLY is only allowed for permanent relations.
+DROP TABLE repack_conc_temp;
+CREATE UNLOGGED TABLE repack_conc_unlogged (i int PRIMARY KEY);
+REPACK (CONCURRENTLY) repack_conc_unlogged;
+ERROR:  cannot repack relation "repack_conc_unlogged"
+HINT:  REPACK CONCURRENTLY is only allowed for permanent relations.
+DROP TABLE repack_conc_unlogged;
+-- Doesn't support TOAST tables directly
+CREATE TABLE repack_conc_toast (t text);
+SELECT reltoastrelid::regclass AS toast_rel
+FROM pg_class WHERE oid = 'repack_conc_toast'::regclass \gset
+\set VERBOSITY sqlstate
+REPACK (CONCURRENTLY) :toast_rel;
+ERROR:  0A000
+\set VERBOSITY default
+DROP TABLE repack_conc_toast;
+-- Doesn't support tables with REPLICA IDENTITY NOTHING, even if they have a primary key
+CREATE TABLE repack_conc_replident (i int PRIMARY KEY);
+ALTER TABLE repack_conc_replident REPLICA IDENTITY NOTHING;
+REPACK (CONCURRENTLY) repack_conc_replident;
+ERROR:  cannot repack relation "repack_conc_replident"
+HINT:  Relation "repack_conc_replident" has insufficient replication identity.
+-- Doesn't support tables with REPLICA IDENTITY FULL, even if they have a primary key
+ALTER TABLE repack_conc_replident REPLICA IDENTITY FULL;
+REPACK (CONCURRENTLY) repack_conc_replident;
+-- Doesn't support tables without a primary key or replica identity index
+ALTER TABLE repack_conc_replident DROP CONSTRAINT repack_conc_replident_pkey;
+ALTER TABLE repack_conc_replident REPLICA IDENTITY DEFAULT;
+REPACK (CONCURRENTLY) repack_conc_replident;
+ERROR:  cannot process relation "repack_conc_replident"
+HINT:  Relation "repack_conc_replident" has no identity index.
+-- Doesn't support tables with deferrable primary keys
+ALTER TABLE repack_conc_replident ADD PRIMARY KEY (i) DEFERRABLE;
+REPACK (CONCURRENTLY) repack_conc_replident;
+ERROR:  cannot process relation "repack_conc_replident"
+HINT:  Relation "repack_conc_replident" has no identity index.
 -- clean up
+DROP TABLE repack_conc_replident;
 DROP TABLE clustertest;
 DROP TABLE clstr_1;
 DROP TABLE clstr_2;
diff --git a/src/test/regress/sql/cluster.sql b/src/test/regress/sql/cluster.sql
index 6b3219bab94..a8cb7ca982d 100644
--- a/src/test/regress/sql/cluster.sql
+++ b/src/test/regress/sql/cluster.sql
@@ -383,13 +383,56 @@ JOIN relnodes_new n ON o.relname = n.relname
 WHERE o.relfilenode <> n.relfilenode
 ORDER BY o.relname;
 
--- concurrently disallowed in catalogs
+--
+-- Check concurrent mode requirements
+--
+
+-- Disallowed in catalogs
 REPACK (CONCURRENTLY) pg_class;
 
--- CONCURRENTLY doesn't like partitioned tables
+-- Doesn't like partitioned tables
 REPACK (CONCURRENTLY) clstrpart;
 
+-- Doesn't support catalog tables
+REPACK (CONCURRENTLY) pg_class;
+
+-- Only support permanent tables, temp and unlogged tables are not supported
+CREATE TEMP TABLE repack_conc_temp (i int PRIMARY KEY);
+REPACK (CONCURRENTLY) repack_conc_temp;
+DROP TABLE repack_conc_temp;
+CREATE UNLOGGED TABLE repack_conc_unlogged (i int PRIMARY KEY);
+REPACK (CONCURRENTLY) repack_conc_unlogged;
+DROP TABLE repack_conc_unlogged;
+
+-- Doesn't support TOAST tables directly
+CREATE TABLE repack_conc_toast (t text);
+SELECT reltoastrelid::regclass AS toast_rel
+FROM pg_class WHERE oid = 'repack_conc_toast'::regclass \gset
+\set VERBOSITY sqlstate
+REPACK (CONCURRENTLY) :toast_rel;
+\set VERBOSITY default
+DROP TABLE repack_conc_toast;
+
+-- Doesn't support tables with REPLICA IDENTITY NOTHING, even if they have a primary key
+CREATE TABLE repack_conc_replident (i int PRIMARY KEY);
+ALTER TABLE repack_conc_replident REPLICA IDENTITY NOTHING;
+REPACK (CONCURRENTLY) repack_conc_replident;
+
+-- Doesn't support tables with REPLICA IDENTITY FULL, even if they have a primary key
+ALTER TABLE repack_conc_replident REPLICA IDENTITY FULL;
+REPACK (CONCURRENTLY) repack_conc_replident;
+
+-- Doesn't support tables without a primary key or replica identity index
+ALTER TABLE repack_conc_replident DROP CONSTRAINT repack_conc_replident_pkey;
+ALTER TABLE repack_conc_replident REPLICA IDENTITY DEFAULT;
+REPACK (CONCURRENTLY) repack_conc_replident;
+
+-- Doesn't support tables with deferrable primary keys
+ALTER TABLE repack_conc_replident ADD PRIMARY KEY (i) DEFERRABLE;
+REPACK (CONCURRENTLY) repack_conc_replident;
+
 -- clean up
+DROP TABLE repack_conc_replident;
 DROP TABLE clustertest;
 DROP TABLE clstr_1;
 DROP TABLE clstr_2;