diff --git a/xarray/backends/api.py b/xarray/backends/api.py
index 9f45474e7e7..c2c655b0893 100644
--- a/xarray/backends/api.py
+++ b/xarray/backends/api.py
@@ -736,6 +736,7 @@ def open_mfdataset(
     parallel=False,
     join="outer",
     attrs_file=None,
+    tolerance=0,
     **kwargs,
 ):
     """Open multiple files as a single dataset.
@@ -849,6 +850,10 @@ def open_mfdataset(
         Path of the file used to read global attributes from.
         By default global attributes are read from the first file provided,
         with wildcard matches sorted by filename.
+    tolerance: numerical or dict of numerical
+        Value used to check equality with numerical tolerance between the
+        coordinates with the same name across all datasets.  If a dict, maps
+        coordinate names to tolerance values.
     **kwargs : optional
         Additional arguments passed on to :py:func:`xarray.open_dataset`.
 
@@ -950,6 +955,7 @@ def open_mfdataset(
                 coords=coords,
                 join=join,
                 combine_attrs="drop",
+                tolerance=tolerance
             )
         else:
             raise ValueError(
diff --git a/xarray/core/combine.py b/xarray/core/combine.py
index 5b3a8bef6a5..6f9fa4a6384 100644
--- a/xarray/core/combine.py
+++ b/xarray/core/combine.py
@@ -1,6 +1,7 @@
 import itertools
 from collections import Counter
 
+import numpy as np
 import pandas as pd
 
 from . import dtypes
@@ -44,7 +45,7 @@ def _infer_tile_ids_from_nested_list(entry, current_pos):
         yield current_pos, entry
 
 
-def _infer_concat_order_from_coords(datasets):
+def _infer_concat_order_from_coords(datasets, tolerance=0):
 
     concat_dims = []
     tile_ids = [() for ds in datasets]
@@ -64,9 +65,25 @@ def _infer_concat_order_from_coords(datasets):
                     "inferring concatenation order"
                 )
 
+            # tolerance can be a dict with dims as key or a constant value
+            if isinstance(tolerance, dict):
+                atol = tolerance.get(dim, 0)
+            else:
+                atol = tolerance
+
             # If dimension coordinate values are same on every dataset then
             # should be leaving this dimension alone (it's just a "bystander")
-            if not all(index.equals(indexes[0]) for index in indexes[1:]):
+            if not (
+                all(index.equals(indexes[0]) for index in indexes[1:])
+                or (
+                    atol > 0
+                    and all(index.is_numeric() for index in indexes)
+                    and all(
+                        np.allclose(index, indexes[0], atol=atol, rtol=0)
+                        for index in indexes[1:]
+                    )
+                )
+            ):
 
                 # Infer order datasets should be arranged in along this dim
                 concat_dims.append(dim)
@@ -547,6 +564,7 @@ def combine_by_coords(
     fill_value=dtypes.NA,
     join="outer",
     combine_attrs="no_conflicts",
+    tolerance=0,
 ):
     """
     Attempt to auto-magically combine the given datasets into one by using
@@ -634,6 +652,10 @@ def combine_by_coords(
           the same name must also have the same value.
         - "override": skip comparing and copy attrs from the first dataset to
           the result.
+    tolerance: numerical or dict of numerical
+        Value used to check equality with numerical tolerance between the
+        coordinates with the same name across all datasets.  If a dict, maps
+        coordinate names to tolerance values.
 
     Returns
     -------
@@ -757,7 +779,7 @@ def combine_by_coords(
     concatenated_grouped_by_data_vars = []
     for vars, datasets_with_same_vars in grouped_by_vars:
         combined_ids, concat_dims = _infer_concat_order_from_coords(
-            list(datasets_with_same_vars)
+            list(datasets_with_same_vars), tolerance
         )
 
         if fill_value is None:
diff --git a/xarray/tests/test_combine.py b/xarray/tests/test_combine.py
index fa1b749b1a7..53f9888a5f6 100644
--- a/xarray/tests/test_combine.py
+++ b/xarray/tests/test_combine.py
@@ -818,6 +818,31 @@ def test_combine_by_coords_incomplete_hypercube(self):
         with pytest.raises(ValueError):
             combine_by_coords([x1, x2, x3], fill_value=None)
 
+    def test_combine_by_coords_with_tolerance(self):
+        x = [0, 1, 2]
+        tol = 1e-7
+
+        x1 = x + tol * np.array([-0.5, 0, 0.5])
+        ds1 = Dataset(
+            {"a": (("time", "x"), [[9, 0, 2]])}, coords={"x": x1, "time": [0]}
+        )
+
+        x2 = x + tol * np.array([+0.4, 0.3, 0.1])
+        ds2 = Dataset(
+            {"a": (("time", "x"), [[6, 8, 3]])}, coords={"x": x2, "time": [1]}
+        )
+
+        # fail if tolerance is not properly implemented
+        combined = combine_by_coords([ds1, ds2], tolerance=tol)
+        # check that x1 was chosen
+        print(combined.x)
+        assert np.all(combined.x == x1)
+        assert len(combined.time) == 2
+
+        # fail if tolerance is not properly implemented
+        combined = combine_by_coords([ds1, ds2], tolerance={"x": tol})
+        assert np.all(combined.x == x1)
+        assert len(combined.time) == 2
 
 @requires_cftime
 def test_combine_by_coords_distant_cftime_dates():