diff --git a/xarray/core/dataset.py b/xarray/core/dataset.py
index 706ccbde8c4..19af7f6c3cd 100644
--- a/xarray/core/dataset.py
+++ b/xarray/core/dataset.py
@@ -2853,6 +2853,7 @@ def interp(
         method: str = "linear",
         assume_sorted: bool = False,
         kwargs: Mapping[str, Any] = None,
+        method_non_numeric: str = "nearest",
         **coords_kwargs: Any,
     ) -> "Dataset":
         """Multidimensional interpolation of Dataset.
@@ -2877,6 +2878,9 @@ def interp(
             Additional keyword arguments passed to scipy's interpolator. Valid
             options and their behavior depend on if 1-dimensional or
             multi-dimensional interpolation is used.
+        method_non_numeric : {"nearest", "pad", "ffill", "backfill", "bfill"}, optional
+            Method for non-numeric types. Passed on to :py:meth:`Dataset.reindex`.
+            ``"nearest"`` is used by default.
         **coords_kwargs : {dim: coordinate, ...}, optional
             The keyword arguments form of ``coords``.
             One of coords or coords_kwargs must be provided.
@@ -3034,6 +3038,7 @@ def _validate_interp_indexer(x, new_x):
             }
 
         variables: Dict[Hashable, Variable] = {}
+        to_reindex: Dict[Hashable, Variable] = {}
         for name, var in obj._variables.items():
             if name in indexers:
                 continue
@@ -3043,20 +3048,45 @@ def _validate_interp_indexer(x, new_x):
             else:
                 use_indexers = validated_indexers
 
-            if var.dtype.kind in "uifc":
+            dtype_kind = var.dtype.kind
+            if dtype_kind in "uifc":
+                # For normal number types do the interpolation:
                 var_indexers = {k: v for k, v in use_indexers.items() if k in var.dims}
                 variables[name] = missing.interp(var, var_indexers, method, **kwargs)
+            elif dtype_kind in "ObU" and (use_indexers.keys() & var.dims):
+                # For types that we do not understand do stepwise
+                # interpolation to avoid modifying the elements.
+                # Use reindex_variables instead because it supports
+                # booleans and objects and retains the dtype but inside
+                # this loop there might be some duplicate code that slows it
+                # down, therefore collect these signals and run it later:
+                to_reindex[name] = var
             elif all(d not in indexers for d in var.dims):
-                # keep unrelated object array
+                # For anything else we can only keep variables if they
+                # are not dependent on any coords that are being
+                # interpolated along:
                 variables[name] = var
 
+        if to_reindex:
+            # Reindex variables:
+            variables_reindex = alignment.reindex_variables(
+                variables=to_reindex,
+                sizes=obj.sizes,
+                indexes=obj.xindexes,
+                indexers={k: v[-1] for k, v in validated_indexers.items()},
+                method=method_non_numeric,
+            )[0]
+            variables.update(variables_reindex)
+
+        # Get the coords that also exist in the variables:
         coord_names = obj._coord_names & variables.keys()
+        # Get the indexes that are not being interpolated along:
         indexes = {k: v for k, v in obj.xindexes.items() if k not in indexers}
         selected = self._replace_with_new_dims(
             variables.copy(), coord_names, indexes=indexes
         )
 
-        # attach indexer as coordinate
+        # Attach indexer as coordinate
         variables.update(indexers)
         for k, v in indexers.items():
             assert isinstance(v, Variable)
@@ -3077,6 +3107,7 @@ def interp_like(
         method: str = "linear",
         assume_sorted: bool = False,
         kwargs: Mapping[str, Any] = None,
+        method_non_numeric: str = "nearest",
     ) -> "Dataset":
         """Interpolate this object onto the coordinates of another object,
         filling the out of range values with NaN.
@@ -3098,6 +3129,9 @@ def interp_like(
             values.
         kwargs : dict, optional
             Additional keyword passed to scipy's interpolator.
+        method_non_numeric : {"nearest", "pad", "ffill", "backfill", "bfill"}, optional
+            Method for non-numeric types. Passed on to :py:meth:`Dataset.reindex`.
+            ``"nearest"`` is used by default.
 
         Returns
         -------
@@ -3133,7 +3167,13 @@ def interp_like(
             # We do not support interpolation along object coordinate.
             # reindex instead.
             ds = self.reindex(object_coords)
-        return ds.interp(numeric_coords, method, assume_sorted, kwargs)
+        return ds.interp(
+            coords=numeric_coords,
+            method=method,
+            assume_sorted=assume_sorted,
+            kwargs=kwargs,
+            method_non_numeric=method_non_numeric,
+        )
 
     # Helper methods for rename()
     def _rename_vars(self, name_dict, dims_dict):
diff --git a/xarray/tests/test_interp.py b/xarray/tests/test_interp.py
index 97a6d236f0a..ab023dc1558 100644
--- a/xarray/tests/test_interp.py
+++ b/xarray/tests/test_interp.py
@@ -416,15 +416,19 @@ def test_errors(use_dask):
 
 @requires_scipy
 def test_dtype():
-    ds = xr.Dataset(
-        {"var1": ("x", [0, 1, 2]), "var2": ("x", ["a", "b", "c"])},
-        coords={"x": [0.1, 0.2, 0.3], "z": ("x", ["a", "b", "c"])},
-    )
-    actual = ds.interp(x=[0.15, 0.25])
-    assert "var1" in actual
-    assert "var2" not in actual
-    # object array should be dropped
-    assert "z" not in actual.coords
+    data_vars = dict(
+        a=("time", np.array([1, 1.25, 2])),
+        b=("time", np.array([True, True, False], dtype=bool)),
+        c=("time", np.array(["start", "start", "end"], dtype=str)),
+    )
+    time = np.array([0, 0.25, 1], dtype=float)
+    expected = xr.Dataset(data_vars, coords=dict(time=time))
+    actual = xr.Dataset(
+        {k: (dim, arr[[0, -1]]) for k, (dim, arr) in data_vars.items()},
+        coords=dict(time=time[[0, -1]]),
+    )
+    actual = actual.interp(time=time, method="linear")
+    assert_identical(expected, actual)
 
 
 @requires_scipy