diff --git a/auditwheel/wheel_abi.py b/auditwheel/wheel_abi.py
index 160b9573..ba8fbee2 100644
--- a/auditwheel/wheel_abi.py
+++ b/auditwheel/wheel_abi.py
@@ -118,10 +118,16 @@ def get_wheel_elfdata(wheel_fn: str):
             # we should walk its elftree.
             if basename(fn) not in needed_libs:
                 full_elftree[fn] = nonpy_elftree[fn]
-                full_external_refs[fn] = lddtree_external_references(
-                    nonpy_elftree[fn], ctx.path)
 
-    log.debug(json.dumps(full_elftree, indent=4))
+            # Even if a non-pyextension ELF file is not needed, we
+            # should include it as an external references, because
+            # they might also require external libraries.
+            full_external_refs[fn] = lddtree_external_references(nonpy_elftree[fn],
+                                                                 ctx.path)
+
+    log.debug('full_elftree:\n%s', json.dumps(full_elftree, indent=4))
+    log.debug('full_external_refs (will be repaired):\n%s',
+        json.dumps(full_external_refs, indent=4))
 
     return (full_elftree, full_external_refs, versioned_symbols,
             uses_ucs2_symbols, uses_PyFPE_jbuf)