diff --git a/docs/source/distributions.rst b/docs/source/distributions.rst
index ec58a174a3..b8ed1c6d93 100644
--- a/docs/source/distributions.rst
+++ b/docs/source/distributions.rst
@@ -529,6 +529,13 @@ SoftplusTransform
     :undoc-members:
     :show-inheritance:
 
+UnitLowerCholeskyTransform
+--------------------------
+.. autoclass:: pyro.distributions.transforms.UnitLowerCholeskyTransform
+    :members:
+    :undoc-members:
+    :show-inheritance:
+
 TransformModules
 ~~~~~~~~~~~~~~~~
 
diff --git a/pyro/distributions/constraints.py b/pyro/distributions/constraints.py
index d6bd399459..711b295cc2 100644
--- a/pyro/distributions/constraints.py
+++ b/pyro/distributions/constraints.py
@@ -101,6 +101,23 @@ class _SoftplusLowerCholesky(type(lower_cholesky)):
     pass
 
 
+class _UnitLowerCholesky(Constraint):
+    """
+    Constrain to lower-triangular square matrices with all ones diagonals.
+    """
+
+    event_dim = 2
+
+    def check(self, value):
+        value_tril = value.tril()
+        lower_triangular = (
+            (value_tril == value).view(value.shape[:-2] + (-1,)).min(-1)[0]
+        )
+
+        ones_diagonal = (value.diagonal(dim1=-2, dim2=-1) == 1).min(-1)[0]
+        return lower_triangular & ones_diagonal
+
+
 corr_matrix = _CorrMatrix()
 integer = _Integer()
 ordered_vector = _OrderedVector()
@@ -108,6 +125,7 @@ class _SoftplusLowerCholesky(type(lower_cholesky)):
 sphere = _Sphere()
 softplus_positive = _SoftplusPositive()
 softplus_lower_cholesky = _SoftplusLowerCholesky()
+unit_lower_cholesky = _UnitLowerCholesky()
 corr_cholesky_constraint = corr_cholesky  # noqa: F405 DEPRECATED
 
 __all__ = [
@@ -119,6 +137,7 @@ class _SoftplusLowerCholesky(type(lower_cholesky)):
     "softplus_lower_cholesky",
     "softplus_positive",
     "sphere",
+    "unit_lower_cholesky",
 ]
 
 __all__.extend(torch_constraints)
diff --git a/pyro/distributions/transforms/__init__.py b/pyro/distributions/transforms/__init__.py
index 298c9c2ade..917da4cffb 100644
--- a/pyro/distributions/transforms/__init__.py
+++ b/pyro/distributions/transforms/__init__.py
@@ -79,6 +79,7 @@
 )
 from .spline_coupling import SplineCoupling, spline_coupling
 from .sylvester import Sylvester, sylvester
+from .unit_cholesky import UnitLowerCholeskyTransform
 
 ########################################
 # register transforms
@@ -132,6 +133,11 @@ def _transform_to_softplus_lower_cholesky(constraint):
     return SoftplusLowerCholeskyTransform()
 
 
+@transform_to.register(constraints.unit_lower_cholesky)
+def _transform_to_unit_lower_cholesky(constraint):
+    return UnitLowerCholeskyTransform()
+
+
 def iterated(repeats, base_fn, *args, **kwargs):
     """
     Helper function to compose a sequence of bijective transforms with potentially
diff --git a/pyro/distributions/transforms/unit_cholesky.py b/pyro/distributions/transforms/unit_cholesky.py
new file mode 100644
index 0000000000..8262f858fb
--- /dev/null
+++ b/pyro/distributions/transforms/unit_cholesky.py
@@ -0,0 +1,32 @@
+# Copyright Contributors to the Pyro project.
+# SPDX-License-Identifier: Apache-2.0
+
+import torch
+from torch.distributions import constraints
+from torch.distributions.transforms import Transform
+
+from pyro.distributions.constraints import unit_lower_cholesky
+
+
+class UnitLowerCholeskyTransform(Transform):
+    """
+    Transform from unconstrained matrices to lower-triangular matrices with
+    all ones diagonals.
+    """
+
+    domain = constraints.independent(constraints.real, 2)
+    codomain = unit_lower_cholesky
+
+    def __eq__(self, other):
+        return isinstance(other, UnitLowerCholeskyTransform)
+
+    def _call(self, x):
+        return x.tril(-1) + torch.eye(x.size(-1), device=x.device, dtype=x.dtype)
+
+    def _inverse(self, y):
+        return y
+
+
+__all__ = [
+    "UnitLowerCholeskyTransform",
+]
diff --git a/pyro/infer/autoguide/guides.py b/pyro/infer/autoguide/guides.py
index 467e9eb003..292e3bcd19 100644
--- a/pyro/infer/autoguide/guides.py
+++ b/pyro/infer/autoguide/guides.py
@@ -863,7 +863,7 @@ class AutoMultivariateNormal(AutoContinuous):
     """
 
     scale_constraint = constraints.softplus_positive
-    scale_tril_constraint = constraints.softplus_lower_cholesky
+    scale_tril_constraint = constraints.unit_lower_cholesky
 
     def __init__(self, model, init_loc_fn=init_to_median, init_scale=0.1):
         if not isinstance(init_scale, float) or not (init_scale > 0):
diff --git a/tests/distributions/test_transforms.py b/tests/distributions/test_transforms.py
index 4344eb56d2..560d3a189e 100644
--- a/tests/distributions/test_transforms.py
+++ b/tests/distributions/test_transforms.py
@@ -456,6 +456,7 @@ def transform_to_vec(x_vec):
     [
         T.LowerCholeskyTransform(),
         T.SoftplusLowerCholeskyTransform(),
+        T.UnitLowerCholeskyTransform(),
     ],
     ids=lambda t: type(t).__name__,
 )