diff --git a/Makefile b/Makefile
index 1e31acc72e..bd8103076a 100644
--- a/Makefile
+++ b/Makefile
@@ -409,6 +409,9 @@ include shared.mak
 # to the "<name>" of the corresponding `compat/fsmonitor/fsm-settings-<name>.c`
 # that implements the `fsm_os_settings__*()` routines.
 #
+# Define LINK_FUZZ_PROGRAMS if you want `make all` to also build the fuzz test
+# programs in oss-fuzz/.
+#
 # === Optional library: libintl ===
 #
 # Define NO_GETTEXT if you don't want Git output to be translated.
@@ -752,23 +755,6 @@ SCRIPTS = $(SCRIPT_SH_GEN) \
 
 ETAGS_TARGET = TAGS
 
-# If you add a new fuzzer, please also make sure to run it in
-# ci/run-build-and-minimal-fuzzers.sh so that we make sure it still links and
-# runs in the future.
-FUZZ_OBJS += oss-fuzz/dummy-cmd-main.o
-FUZZ_OBJS += oss-fuzz/fuzz-commit-graph.o
-FUZZ_OBJS += oss-fuzz/fuzz-config.o
-FUZZ_OBJS += oss-fuzz/fuzz-date.o
-FUZZ_OBJS += oss-fuzz/fuzz-pack-headers.o
-FUZZ_OBJS += oss-fuzz/fuzz-pack-idx.o
-.PHONY: fuzz-objs
-fuzz-objs: $(FUZZ_OBJS)
-
-# Always build fuzz objects even if not testing, to prevent bit-rot.
-all:: $(FUZZ_OBJS)
-
-FUZZ_PROGRAMS += $(patsubst %.o,%,$(filter-out %dummy-cmd-main.o,$(FUZZ_OBJS)))
-
 # Empty...
 EXTRA_PROGRAMS =
 
@@ -2373,6 +2359,29 @@ ifndef NO_TCLTK
 endif
 	$(QUIET_SUBDIR0)templates $(QUIET_SUBDIR1) SHELL_PATH='$(SHELL_PATH_SQ)' PERL_PATH='$(PERL_PATH_SQ)'
 
+# If you add a new fuzzer, please also make sure to run it in
+# ci/run-build-and-minimal-fuzzers.sh so that we make sure it still links and
+# runs in the future.
+FUZZ_OBJS += oss-fuzz/dummy-cmd-main.o
+FUZZ_OBJS += oss-fuzz/fuzz-commit-graph.o
+FUZZ_OBJS += oss-fuzz/fuzz-config.o
+FUZZ_OBJS += oss-fuzz/fuzz-date.o
+FUZZ_OBJS += oss-fuzz/fuzz-pack-headers.o
+FUZZ_OBJS += oss-fuzz/fuzz-pack-idx.o
+.PHONY: fuzz-objs
+fuzz-objs: $(FUZZ_OBJS)
+
+# Always build fuzz objects even if not testing, to prevent bit-rot.
+all:: $(FUZZ_OBJS)
+
+FUZZ_PROGRAMS += $(patsubst %.o,%,$(filter-out %dummy-cmd-main.o,$(FUZZ_OBJS)))
+
+# Build fuzz programs when possible, even without the necessary fuzzing support,
+# to prevent bit-rot.
+ifdef LINK_FUZZ_PROGRAMS
+all:: $(FUZZ_PROGRAMS)
+endif
+
 please_set_SHELL_PATH_to_a_more_modern_shell:
 	@$$(:)
 
@@ -3858,22 +3867,22 @@ cover_db_html: cover_db
 #
 # An example command to build against libFuzzer from LLVM 11.0.0:
 #
-# make CC=clang CXX=clang++ \
+# make CC=clang FUZZ_CXX=clang++ \
 #      CFLAGS="-fsanitize=fuzzer-no-link,address" \
 #      LIB_FUZZING_ENGINE="-fsanitize=fuzzer,address" \
 #      fuzz-all
 #
+FUZZ_CXX ?= $(CC)
 FUZZ_CXXFLAGS ?= $(ALL_CFLAGS)
 
 .PHONY: fuzz-all
+fuzz-all: $(FUZZ_PROGRAMS)
 
 $(FUZZ_PROGRAMS): %: %.o oss-fuzz/dummy-cmd-main.o $(GITLIBS) GIT-LDFLAGS
-	$(QUIET_LINK)$(CXX) $(FUZZ_CXXFLAGS) -o $@ $(ALL_LDFLAGS) \
+	$(QUIET_LINK)$(FUZZ_CXX) $(FUZZ_CXXFLAGS) -o $@ $(ALL_LDFLAGS) \
 		-Wl,--allow-multiple-definition \
 		$(filter %.o,$^) $(filter %.a,$^) $(LIBS) $(LIB_FUZZING_ENGINE)
 
-fuzz-all: $(FUZZ_PROGRAMS)
-
 $(UNIT_TEST_PROGS): $(UNIT_TEST_BIN)/%$X: $(UNIT_TEST_DIR)/%.o $(UNIT_TEST_DIR)/test-lib.o $(GITLIBS) GIT-LDFLAGS
 	$(call mkdir_p_parent_template)
 	$(QUIET_LINK)$(CC) $(ALL_CFLAGS) -o $@ $(ALL_LDFLAGS) \
diff --git a/ci/run-build-and-minimal-fuzzers.sh b/ci/run-build-and-minimal-fuzzers.sh
index a51076d18d..797d65c661 100755
--- a/ci/run-build-and-minimal-fuzzers.sh
+++ b/ci/run-build-and-minimal-fuzzers.sh
@@ -7,7 +7,7 @@
 
 group "Build fuzzers" make \
 	CC=clang \
-	CXX=clang++ \
+	FUZZ_CXX=clang++ \
 	CFLAGS="-fsanitize=fuzzer-no-link,address" \
 	LIB_FUZZING_ENGINE="-fsanitize=fuzzer,address" \
 	fuzz-all
diff --git a/config.mak.uname b/config.mak.uname
index a7607a5676..85d63821ec 100644
--- a/config.mak.uname
+++ b/config.mak.uname
@@ -68,6 +68,7 @@ ifeq ($(uname_S),Linux)
         ifneq ($(findstring .el7.,$(uname_R)),)
 		BASIC_CFLAGS += -std=c99
         endif
+	LINK_FUZZ_PROGRAMS = YesPlease
 endif
 ifeq ($(uname_S),GNU/kFreeBSD)
 	HAVE_ALLOCA_H = YesPlease