diff --git a/elf/Makefile b/elf/Makefile
index 3386f0ce77..3928a08787 100644
--- a/elf/Makefile
+++ b/elf/Makefile
@@ -374,6 +374,8 @@ tests += \
   tst-align \
   tst-align2 \
   tst-align3 \
+  tst-audit-tlsdesc \
+  tst-audit-tlsdesc-dlopen \
   tst-audit1 \
   tst-audit2 \
   tst-audit8 \
@@ -765,6 +767,8 @@ modules-names += \
   tst-alignmod3 \
   tst-array2dep \
   tst-array5dep \
+  tst-audit-tlsdesc-mod1 \
+  tst-audit-tlsdesc-mod2 \
   tst-audit11mod1 \
   tst-audit11mod2 \
   tst-audit12mod1 \
@@ -798,6 +802,7 @@ modules-names += \
   tst-auditmanymod7 \
   tst-auditmanymod8 \
   tst-auditmanymod9 \
+  tst-auditmod-tlsdesc  \
   tst-auditmod1 \
   tst-auditmod9a \
   tst-auditmod9b \
@@ -990,23 +995,8 @@ modules-names += tst-gnu2-tls1mod
 $(objpfx)tst-gnu2-tls1: $(objpfx)tst-gnu2-tls1mod.so
 tst-gnu2-tls1mod.so-no-z-defs = yes
 CFLAGS-tst-gnu2-tls1mod.c += -mtls-dialect=gnu2
+endif # $(have-mtls-dialect-gnu2)
 
-tests += tst-audit-tlsdesc tst-audit-tlsdesc-dlopen
-modules-names += tst-audit-tlsdesc-mod1 tst-audit-tlsdesc-mod2 tst-auditmod-tlsdesc
-$(objpfx)tst-audit-tlsdesc: $(objpfx)tst-audit-tlsdesc-mod1.so \
-			    $(objpfx)tst-audit-tlsdesc-mod2.so \
-			    $(shared-thread-library)
-CFLAGS-tst-audit-tlsdesc-mod1.c += -mtls-dialect=gnu2
-CFLAGS-tst-audit-tlsdesc-mod2.c += -mtls-dialect=gnu2
-$(objpfx)tst-audit-tlsdesc-dlopen: $(shared-thread-library)
-$(objpfx)tst-audit-tlsdesc-dlopen.out: $(objpfx)tst-audit-tlsdesc-mod1.so \
-				       $(objpfx)tst-audit-tlsdesc-mod2.so
-$(objpfx)tst-audit-tlsdesc-mod1.so: $(objpfx)tst-audit-tlsdesc-mod2.so
-$(objpfx)tst-audit-tlsdesc.out: $(objpfx)tst-auditmod-tlsdesc.so
-tst-audit-tlsdesc-ENV = LD_AUDIT=$(objpfx)tst-auditmod-tlsdesc.so
-$(objpfx)tst-audit-tlsdesc-dlopen.out: $(objpfx)tst-auditmod-tlsdesc.so
-tst-audit-tlsdesc-dlopen-ENV = LD_AUDIT=$(objpfx)tst-auditmod-tlsdesc.so
-endif
 ifeq (yes,$(have-protected-data))
 modules-names += tst-protected1moda tst-protected1modb
 tests += tst-protected1a tst-protected1b
@@ -2968,3 +2958,21 @@ $(objpfx)tst-tls-allocation-failure-static-patched.out: \
 	grep -q '^Fatal glibc error: Cannot allocate TLS block$$' $@ \
 	  && grep -q '^status: 127$$' $@; \
 	  $(evaluate-test)
+
+$(objpfx)tst-audit-tlsdesc: $(objpfx)tst-audit-tlsdesc-mod1.so \
+			    $(objpfx)tst-audit-tlsdesc-mod2.so \
+			    $(shared-thread-library)
+ifeq (yes,$(have-mtls-dialect-gnu2))
+# The test is valid for all TLS types, but we want to exercise GNU2
+# TLS if possible.
+CFLAGS-tst-audit-tlsdesc-mod1.c += -mtls-dialect=gnu2
+CFLAGS-tst-audit-tlsdesc-mod2.c += -mtls-dialect=gnu2
+endif
+$(objpfx)tst-audit-tlsdesc-dlopen: $(shared-thread-library)
+$(objpfx)tst-audit-tlsdesc-dlopen.out: $(objpfx)tst-audit-tlsdesc-mod1.so \
+				       $(objpfx)tst-audit-tlsdesc-mod2.so
+$(objpfx)tst-audit-tlsdesc-mod1.so: $(objpfx)tst-audit-tlsdesc-mod2.so
+$(objpfx)tst-audit-tlsdesc.out: $(objpfx)tst-auditmod-tlsdesc.so
+tst-audit-tlsdesc-ENV = LD_AUDIT=$(objpfx)tst-auditmod-tlsdesc.so
+$(objpfx)tst-audit-tlsdesc-dlopen.out: $(objpfx)tst-auditmod-tlsdesc.so
+tst-audit-tlsdesc-dlopen-ENV = LD_AUDIT=$(objpfx)tst-auditmod-tlsdesc.so