diff --git a/core/src/main/java/org/apache/calcite/rel/rel2sql/SqlImplementor.java b/core/src/main/java/org/apache/calcite/rel/rel2sql/SqlImplementor.java
index 52266cb6f9ff..41418e100f2a 100644
--- a/core/src/main/java/org/apache/calcite/rel/rel2sql/SqlImplementor.java
+++ b/core/src/main/java/org/apache/calcite/rel/rel2sql/SqlImplementor.java
@@ -1751,7 +1751,7 @@ private Builder builder(RelNode rel, Set<Clause> clauses) {
                 if (selectItem.i != ordinal) {
                   final @Nullable String alias =
                       SqlValidatorUtil.alias(selectItem.e);
-                  if (name.equalsIgnoreCase(alias)) {
+                  if (name.equalsIgnoreCase(alias) && dialect.getConformance().isSortByAlias()) {
                     return SqlLiteral.createExactNumeric(
                         Integer.toString(ordinal + 1), SqlParserPos.ZERO);
                   }
@@ -1818,9 +1818,10 @@ private boolean needNewSubQuery(
 
       if (rel instanceof Project
           && clauses.contains(Clause.ORDER_BY)
-          && dialect.getConformance().isSortByOrdinal()) {
+          && dialect.getConformance().isSortByOrdinal()
+          && hasSortByOrdinal()) {
         // Cannot merge a Project that contains sort by ordinal under it.
-        return hasSortByOrdinal();
+        return true;
       }
 
       if (rel instanceof Aggregate) {
@@ -1855,7 +1856,7 @@ private boolean hasSortByOrdinal(@UnknownInitialization Result this) {
         }
         for (SqlNode sqlNode : orderList) {
           if (!(sqlNode instanceof SqlBasicCall)) {
-            return false;
+            return sqlNode instanceof SqlNumericLiteral;
           }
           for (SqlNode operand : ((SqlBasicCall) sqlNode).getOperandList()) {
             if (operand instanceof SqlNumericLiteral) {
diff --git a/core/src/test/java/org/apache/calcite/rel/rel2sql/RelToSqlConverterTest.java b/core/src/test/java/org/apache/calcite/rel/rel2sql/RelToSqlConverterTest.java
index 7016a74b84e0..c690203affd5 100644
--- a/core/src/test/java/org/apache/calcite/rel/rel2sql/RelToSqlConverterTest.java
+++ b/core/src/test/java/org/apache/calcite/rel/rel2sql/RelToSqlConverterTest.java
@@ -879,6 +879,78 @@ private static String toSql(RelNode root, SqlDialect dialect,
         .withPresto().ok(expectedPresto);
   }
 
+  /**
+   * Test case for
+   * <a href="https://issues.apache.org/jira/browse/CALCITE-5530">[CALCITE-5530]
+   * RelToSqlConverter[ORDER BY] generates an incorrect field alias
+   * when 2 projection fields have the same name</a>.
+   */
+  @Test void testOrderByFieldNotInTheProjectionWithASameAliasAsThatInTheProjection() {
+    final RelBuilder builder = relBuilder();
+    final RelNode base = builder
+        .scan("EMP")
+        .project(
+            builder.alias(
+                builder.call(SqlStdOperatorTable.UPPER, builder.field("ENAME")), "EMPNO"),
+            builder.field("EMPNO")
+        )
+        .sort(1)
+        .project(builder.field(0))
+        .build();
+
+    // The expected string should deliberately have a subquery to handle a scenario in which
+    // the projection field has an alias with the same name as that of the field used in the
+    // ORDER BY
+    String expectedSql1 = ""
+        + "SELECT \"EMPNO\"\n"
+        + "FROM (SELECT UPPER(\"ENAME\") AS \"EMPNO\", \"EMPNO\" AS \"EMPNO0\"\n"
+        + "FROM \"scott\".\"EMP\"\n"
+        + "ORDER BY 2) AS \"t0\"";
+    String actualSql1 = toSql(base);
+    assertThat(actualSql1, isLinux(expectedSql1));
+
+    String actualSql2 = toSql(base, nonOrdinalDialect());
+    String expectedSql2 = "SELECT UPPER(ENAME) AS EMPNO\n"
+        + "FROM scott.EMP\n"
+        + "ORDER BY EMPNO";
+    assertThat(actualSql2, isLinux(expectedSql2));
+  }
+
+  @Test void testOrderByExpressionNotInTheProjectionThatRefersToUnderlyingFieldWithSameAlias() {
+    final RelBuilder builder = relBuilder();
+    final RelNode base = builder
+        .scan("EMP")
+        .project(
+            builder.alias(
+                builder.call(SqlStdOperatorTable.UPPER, builder.field("ENAME")), "EMPNO"),
+            builder.call(
+                SqlStdOperatorTable.PLUS, builder.field("EMPNO"),
+                builder.literal(1)
+            )
+        )
+        .sort(1)
+        .project(builder.field(0))
+        .build();
+
+    // An output such as
+    // "SELECT UPPER(\"ENAME\") AS \"EMPNO\"\nFROM \"scott\".\"EMP\"\nORDER BY \"EMPNO\" + 1"
+    // would be incorrect since the rel is sorting by the field \"EMPNO\" + 1 in which EMPNO
+    // refers to the physical column EMPNO and not the alias
+    String actualSql1 = toSql(base);
+    String expectedSql1 = ""
+        + "SELECT \"EMPNO\"\n"
+        + "FROM (SELECT UPPER(\"ENAME\") AS \"EMPNO\", \"EMPNO\" + 1 AS \"$f1\"\n"
+        + "FROM \"scott\".\"EMP\"\n"
+        + "ORDER BY 2) AS \"t0\"";
+    assertThat(actualSql1, isLinux(expectedSql1));
+
+    String actualSql2 = toSql(base, nonOrdinalDialect());
+    String expectedSql2 = "SELECT UPPER(ENAME) AS EMPNO\n"
+        + "FROM scott.EMP\n"
+        + "ORDER BY EMPNO + 1";
+    assertThat(actualSql2, isLinux(expectedSql2));
+  }
+
   @Test void testSelectQueryWithMinAggregateFunction() {
     String query = "select min(\"net_weight\") from \"product\" group by \"product_class_id\" ";
     final String expected = "SELECT MIN(\"net_weight\")\n"