diff --git a/tests/integration/nonpy_rpath/README.md b/tests/integration/nonpy_rpath/README.md
new file mode 100644
index 00000000..e01b25e4
--- /dev/null
+++ b/tests/integration/nonpy_rpath/README.md
@@ -0,0 +1,9 @@
+# Python 3 extension with non-Python library dependency
+
+This example was inspired from https://gist.github.com/physacco/2e1b52415f3a964ad2a542a99bebed8f
+
+This test extension builds two libraries: `_nonpy_rpath.*.so` and `lib_cryptexample.*.so`, where the `*` is a string composed of Python ABI versions and platform tags.
+
+The extension `lib_cryptexample.*.so` should be repaired by auditwheel because it is a needed library, even though it is not a Python extension.
+
+[Issue #136](https://github.com/pypa/auditwheel/issues/136) documents the underlying problem that this test case is designed to solve.
diff --git a/tests/integration/nonpy_rpath/extensions/testcrypt.cpp b/tests/integration/nonpy_rpath/extensions/testcrypt.cpp
new file mode 100644
index 00000000..5a3c1e39
--- /dev/null
+++ b/tests/integration/nonpy_rpath/extensions/testcrypt.cpp
@@ -0,0 +1,6 @@
+#include "testcrypt.h"
+#include <crypt.h>
+
+std::string crypt_something() {
+    return std::string(crypt("will error out", NULL));
+}
diff --git a/tests/integration/nonpy_rpath/extensions/testcrypt.h b/tests/integration/nonpy_rpath/extensions/testcrypt.h
new file mode 100644
index 00000000..1fff29d2
--- /dev/null
+++ b/tests/integration/nonpy_rpath/extensions/testcrypt.h
@@ -0,0 +1,4 @@
+#pragma once
+#include <string>
+
+std::string crypt_something(void);
diff --git a/tests/integration/nonpy_rpath/nonpy_rpath.cpp b/tests/integration/nonpy_rpath/nonpy_rpath.cpp
new file mode 100644
index 00000000..49bcbf42
--- /dev/null
+++ b/tests/integration/nonpy_rpath/nonpy_rpath.cpp
@@ -0,0 +1,25 @@
+#define PY_SSIZE_T_CLEAN
+#include <Python.h>
+#include "extensions/testcrypt.h"
+
+// Module method definitions
+static PyObject* crypt_something(PyObject *self, PyObject *args) {
+    return PyUnicode_FromString(crypt_something().c_str());
+}
+
+/* Module initialization */
+PyMODINIT_FUNC PyInit__nonpy_rpath(void)
+{
+    static PyMethodDef module_methods[] = {
+        {"crypt_something", (PyCFunction)crypt_something, METH_NOARGS, "crypt_something."},
+        {NULL}  /* Sentinel */
+    };
+    static struct PyModuleDef moduledef = {
+        PyModuleDef_HEAD_INIT,
+        "_nonpy_rpath",
+        "_nonpy_rpath module",
+        -1,
+        module_methods,
+    };
+    return PyModule_Create(&moduledef);
+}
diff --git a/tests/integration/nonpy_rpath/nonpy_rpath/__init__.py b/tests/integration/nonpy_rpath/nonpy_rpath/__init__.py
new file mode 100644
index 00000000..1779ebb1
--- /dev/null
+++ b/tests/integration/nonpy_rpath/nonpy_rpath/__init__.py
@@ -0,0 +1,3 @@
+from ._nonpy_rpath import crypt_something
+
+__all__ = ["crypt_something"]
diff --git a/tests/integration/nonpy_rpath/setup.py b/tests/integration/nonpy_rpath/setup.py
new file mode 100644
index 00000000..e4f35458
--- /dev/null
+++ b/tests/integration/nonpy_rpath/setup.py
@@ -0,0 +1,98 @@
+import setuptools.command.build_ext
+from setuptools import setup, find_packages, Distribution
+from setuptools.extension import Extension, Library
+import os
+
+# despite its name, setuptools.command.build_ext.link_shared_object won't
+# link a shared object on Linux, but a static library and patches distutils
+# for this ... We're patching this back now.
+
+
+def always_link_shared_object(
+    self,
+    objects,
+    output_libname,
+    output_dir=None,
+    libraries=None,
+    library_dirs=None,
+    runtime_library_dirs=None,
+    export_symbols=None,
+    debug=0,
+    extra_preargs=None,
+    extra_postargs=None,
+    build_temp=None,
+    target_lang=None,
+):
+    self.link(
+        self.SHARED_LIBRARY,
+        objects,
+        output_libname,
+        output_dir,
+        libraries,
+        library_dirs,
+        runtime_library_dirs,
+        export_symbols,
+        debug,
+        extra_preargs,
+        extra_postargs,
+        build_temp,
+        target_lang,
+    )
+
+
+setuptools.command.build_ext.libtype = "shared"
+setuptools.command.build_ext.link_shared_object = always_link_shared_object
+
+libtype = setuptools.command.build_ext.libtype
+build_ext_cmd = Distribution().get_command_obj("build_ext")
+build_ext_cmd.initialize_options()
+build_ext_cmd.setup_shlib_compiler()
+
+
+def libname(name):
+    """ gets 'name' and returns something like libname.cpython-37m-darwin.so"""
+    filename = build_ext_cmd.get_ext_filename(name)
+    fn, ext = os.path.splitext(filename)
+    return build_ext_cmd.shlib_compiler.library_filename(fn, libtype)
+
+
+pkg_name = "nonpy_rpath"
+crypt_name = "_cryptexample"
+crypt_soname = libname(crypt_name)
+
+build_cmd = Distribution().get_command_obj("build")
+build_cmd.finalize_options()
+build_platlib = build_cmd.build_platlib
+
+
+def link_args(soname=None):
+    args = []
+    if soname:
+        args += ["-Wl,-soname," + soname]
+    loader_path = "$ORIGIN"
+    args += ["-Wl,-rpath," + loader_path]
+    return args
+
+
+nonpy_rpath_module = Extension(
+    pkg_name + "._nonpy_rpath",
+    language="c++",
+    sources=["nonpy_rpath.cpp"],
+    extra_link_args=link_args(),
+    extra_objects=[build_platlib + "/nonpy_rpath/" + crypt_soname],
+)
+crypt_example = Library(
+    pkg_name + "." + crypt_name,
+    language="c++",
+    extra_compile_args=["-lcrypt"],
+    extra_link_args=link_args(crypt_soname) + ["-lcrypt"],
+    sources=["extensions/testcrypt.cpp"],
+)
+
+setup(
+    name="nonpy_rpath",
+    version="0.1.0",
+    packages=find_packages(),
+    description="Test package for nonpy_rpath",
+    ext_modules=[crypt_example, nonpy_rpath_module],
+)
diff --git a/tests/integration/test_manylinux.py b/tests/integration/test_manylinux.py
index a14b6430..40e25d22 100644
--- a/tests/integration/test_manylinux.py
+++ b/tests/integration/test_manylinux.py
@@ -219,10 +219,10 @@ def assert_show_output(manylinux_ctr, wheel, expected_tag, strict):
         assert match['tag'] == expected_tag
     else:
         expected_match = TAG_RE.match(expected_tag)
-        assert expected_match
+        assert expected_match, f"No match for tag {expected_tag}"
         expected_glibc = (int(expected_match['major']), int(expected_match['minor']))
         actual_match = TAG_RE.match(match['tag'])
-        assert actual_match
+        assert actual_match, f"No match for tag {match['tag']}"
         actual_glibc = (int(actual_match['major']), int(actual_match['minor']))
         assert expected_match['arch'] == actual_match['arch']
         assert actual_glibc <= expected_glibc
@@ -705,3 +705,35 @@ def test_build_wheel_compat(target_policy, only_plat, any_manylinux_container,
         ['python', '-c',
          'from sys import exit; from testsimple import run; exit(run())']
     )
+
+
+def test_nonpy_rpath(any_manylinux_container, docker_python, io_folder):
+    # Tests https://github.com/pypa/auditwheel/issues/136
+    policy, tag, manylinux_ctr = any_manylinux_container
+    docker_exec(
+        manylinux_ctr,
+        ['bash', '-c', 'cd /auditwheel_src/tests/integration/nonpy_rpath '
+                       '&& python -m pip wheel --no-deps -w /io .']
+    )
+
+    orig_wheel, *_ = os.listdir(io_folder)
+    assert orig_wheel.startswith("nonpy_rpath-0.1.0")
+    assert 'manylinux' not in orig_wheel
+
+    # Repair the wheel using the appropriate manylinux container
+    repair_command = \
+        f'auditwheel repair --plat {policy} --only-plat -w /io /io/{orig_wheel}'
+    docker_exec(manylinux_ctr, repair_command)
+    filenames = os.listdir(io_folder)
+    assert len(filenames) == 2
+    repaired_wheel = f'nonpy_rpath-0.1.0-{PYTHON_ABI}-{tag}.whl'
+    assert repaired_wheel in filenames
+    assert_show_output(manylinux_ctr, repaired_wheel, policy, False)
+
+    # Test the resulting wheel outside the manylinux container
+    docker_exec(docker_python, "pip install /io/" + repaired_wheel)
+    docker_exec(
+        docker_python,
+        ["python", "-c",
+         "import nonpy_rpath; assert nonpy_rpath.crypt_something().startswith('*')"]
+    )