diff --git a/include/pybind11/eigen/matrix.h b/include/pybind11/eigen/matrix.h
index ca599c954c..0773fb9834 100644
--- a/include/pybind11/eigen/matrix.h
+++ b/include/pybind11/eigen/matrix.h
@@ -86,6 +86,9 @@ struct EigenConformable {
     EigenIndex rows = 0, cols = 0;
     EigenDStride stride{0, 0};    // Only valid if negativestrides is false!
     bool negativestrides = false; // If true, do not use stride!
+    // A numpy stride that is not a whole number of scalars (e.g. a field of a packed structured
+    // array) cannot be written as an Eigen stride. If true, do not use stride!
+    bool badstrides = false;
 
     // NOLINTNEXTLINE(google-explicit-constructor)
     EigenConformable(bool fits = false) : conformable{fits} {}
@@ -115,6 +118,9 @@ struct EigenConformable {
         if (rows == 0 || cols == 0) {
             return true;
         }
+        if (badstrides) {
+            return false;
+        }
         return (props::inner_stride == Eigen::Dynamic || props::inner_stride == stride.inner()
                 || (EigenRowMajor ? cols : rows) == 1)
                && (props::outer_stride == Eigen::Dynamic || props::outer_stride == stride.outer()
@@ -185,19 +191,28 @@ struct EigenProps {
                 return false;
             }
 
-            return {np_rows, np_cols, np_rstride, np_cstride};
+            EigenConformable<row_major> result{np_rows, np_cols, np_rstride, np_cstride};
+            result.badstrides = !whole_scalars(a.strides(0)) || !whole_scalars(a.strides(1));
+            return result;
         }
 
         // Otherwise we're storing an n-vector.  Only one of the strides will be used, but
         // whichever is used, we want the (single) numpy stride value.
         const EigenIndex n = a.shape(0),
                          stride = a.strides(0) / static_cast<ssize_t>(sizeof(Scalar));
+        const bool bad = !whole_scalars(a.strides(0));
+
+        auto make = [bad](EigenIndex r, EigenIndex c, EigenIndex s) {
+            EigenConformable<row_major> result{r, c, s};
+            result.badstrides = bad;
+            return result;
+        };
 
         if (vector) { // Eigen type is a compile-time vector
             if (fixed && size != n) {
                 return false; // Vector size mismatch
             }
-            return {rows == 1 ? 1 : n, cols == 1 ? 1 : n, stride};
+            return make(rows == 1 ? 1 : n, cols == 1 ? 1 : n, stride);
         }
         if (fixed) {
             // The type has a fixed size, but is not a vector: abort
@@ -209,12 +224,18 @@ struct EigenProps {
             if (cols != n) {
                 return false;
             }
-            return {1, n, stride};
+            return make(1, n, stride);
         } // Otherwise it's either fully dynamic, or column dynamic; both become a column vector
         if (fixed_rows && rows != n) {
             return false;
         }
-        return {n, 1, stride};
+        return make(n, 1, stride);
+    }
+
+    // True if a numpy byte stride is a whole number of scalars, and can therefore become an Eigen
+    // stride.
+    static bool whole_scalars(ssize_t byte_stride) {
+        return byte_stride % static_cast<ssize_t>(sizeof(Scalar)) == 0;
     }
 
     static constexpr bool show_writeable
diff --git a/tests/test_eigen_matrix.py b/tests/test_eigen_matrix.py
index 9324c2a7d2..09b36c35db 100644
--- a/tests/test_eigen_matrix.py
+++ b/tests/test_eigen_matrix.py
@@ -221,6 +221,45 @@ def test_negative_stride_from_python(msg):
     )
 
 
+def test_stride_not_multiple_of_scalar_from_python():
+    """A field of a packed structured array has a stride that is not a multiple of the scalar
+    size.  Such an array must not be seen as contiguous (see #6159)."""
+
+    structured = np.zeros(3, dtype=[("a", "f4"), ("b", "i1")])
+    structured["a"] = [1.0, 2.0, 3.0]
+    structured["b"] = [7, 8, 9]
+    field = structured["a"]
+    assert field.strides[0] % field.itemsize != 0
+
+    # By value: the array is copied, the neighbouring field stays unchanged.
+    np.testing.assert_array_equal(m.double_col(field), [2.0, 4.0, 6.0])
+    np.testing.assert_array_equal(structured["a"], [1.0, 2.0, 3.0])
+    np.testing.assert_array_equal(structured["b"], [7, 8, 9])
+
+    # A const Ref is also allowed to copy.
+    wide = np.zeros(6, dtype=[("a", "f8"), ("b", "i1")])
+    wide["a"] = np.arange(6.0)
+    assert m.get_elem_direct(wide["a"]) == 5.0
+
+    # A mutable Ref cannot map this layout, so it must be refused.
+    with pytest.raises(TypeError):
+        m.double_threec(field)
+    np.testing.assert_array_equal(structured["a"], [1.0, 2.0, 3.0])
+    np.testing.assert_array_equal(structured["b"], [7, 8, 9])
+
+    mat = np.zeros((2, 3), dtype=[("a", "f8"), ("b", "i1")])
+    mat["a"] = np.arange(6.0).reshape((2, 3))
+    mat["b"] = 1
+    np.testing.assert_array_equal(
+        m.double_mat_rm(mat["a"]), 2.0 * np.arange(6.0).reshape((2, 3))
+    )
+    np.testing.assert_array_equal(m.diagonal(mat["a"]), [0.0, 4.0])
+    with pytest.raises(TypeError):
+        m.add_rm(mat["a"], 0, 0, 100.0)
+    np.testing.assert_array_equal(mat["a"], np.arange(6.0).reshape((2, 3)))
+    np.testing.assert_array_equal(mat["b"], np.ones((2, 3), dtype="i1"))
+
+
 def test_block_runtime_error_type_caster_eigen_ref_made_a_copy():
     with pytest.raises(RuntimeError) as excinfo:
         m.block(ref, 0, 0, 0, 0)