diff --git a/src/bootstrap/cc_detect.rs b/src/bootstrap/cc_detect.rs
index d50e4cf5269..e750c2963dd 100644
--- a/src/bootstrap/cc_detect.rs
+++ b/src/bootstrap/cc_detect.rs
@@ -129,7 +129,8 @@ pub fn find(build: &mut Build) {
             set_compiler(&mut cfg, Language::CPlusPlus, target, config, build);
             true
         } else {
-            false
+            // Use an auto-detected compiler (or one configured via `CXX_target_triple` env vars).
+            cfg.try_get_compiler().is_ok()
         };
 
         // for VxWorks, record CXX compiler which will be used in lib.rs:linker()
diff --git a/src/bootstrap/test.rs b/src/bootstrap/test.rs
index 33e85dc5e2a..6c2c05ac719 100644
--- a/src/bootstrap/test.rs
+++ b/src/bootstrap/test.rs
@@ -1202,7 +1202,7 @@ note: if you're sure you want to do this, please open an issue as to why. In the
 
             // Only pass correct values for these flags for the `run-make` suite as it
             // requires that a C++ compiler was configured which isn't always the case.
-            if !builder.config.dry_run && suite == "run-make-fulldeps" {
+            if !builder.config.dry_run && matches!(suite, "run-make" | "run-make-fulldeps") {
                 cmd.arg("--cc")
                     .arg(builder.cc(target))
                     .arg("--cxx")
diff --git a/src/test/run-make-fulldeps/issue-36710/Makefile b/src/test/run-make/issue-36710/Makefile
similarity index 57%
rename from src/test/run-make-fulldeps/issue-36710/Makefile
rename to src/test/run-make/issue-36710/Makefile
index 4f93d97636e..b7bf366c918 100644
--- a/src/test/run-make-fulldeps/issue-36710/Makefile
+++ b/src/test/run-make/issue-36710/Makefile
@@ -1,6 +1,6 @@
--include ../tools.mk
+include ../../run-make-fulldeps/tools.mk
 
-# ignore-musl
+# ignore-32bit wrong/no cross compiler and sometimes we pass wrong gcc args (-m64)
 
 all: foo
 	$(call RUN,foo)
diff --git a/src/test/run-make-fulldeps/issue-36710/foo.cpp b/src/test/run-make/issue-36710/foo.cpp
similarity index 100%
rename from src/test/run-make-fulldeps/issue-36710/foo.cpp
rename to src/test/run-make/issue-36710/foo.cpp
diff --git a/src/test/run-make-fulldeps/issue-36710/foo.rs b/src/test/run-make/issue-36710/foo.rs
similarity index 100%
rename from src/test/run-make-fulldeps/issue-36710/foo.rs
rename to src/test/run-make/issue-36710/foo.rs