diff --git a/devito/ir/clusters/algorithms.py b/devito/ir/clusters/algorithms.py
index 7f09afd1c0..4223e789e7 100644
--- a/devito/ir/clusters/algorithms.py
+++ b/devito/ir/clusters/algorithms.py
@@ -121,6 +121,13 @@ class Schedule(Queue):
           Dimension in both Clusters.
     """
 
+    FISSION_THRESHOLD = 2
+    """
+    The maximum number of iteration Dimensions such that we consider fissioning
+    a sequence of Clusters to increase parallelism. IOW, if there are more than
+    this number of iteration Dimensions, we do not even try to fission.
+    """
+
     @timed_pass(name='schedule')
     def process(self, clusters):
         return self._process_fatd(clusters, 1)
@@ -134,7 +141,8 @@ def callback(self, clusters, prefix, backlog=None, known_break=None):
 
         # Take the innermost Dimension -- no other Clusters other than those in
         # `clusters` are supposed to share it
-        candidates = prefix[-1].dim._defines
+        dim = prefix[-1].dim
+        candidates = dim._defines
 
         scope = Scope(flatten(c.exprs for c in clusters))
 
@@ -157,7 +165,7 @@ def callback(self, clusters, prefix, backlog=None, known_break=None):
         # Schedule Clusters over different IterationSpaces if this increases
         # parallelism
         for i in range(1, len(clusters)):
-            if self._break_for_parallelism(scope, candidates, i):
+            if self._break_for_parallelism(scope, dim, i):
                 return self.callback(clusters[:i], prefix, clusters[i:] + backlog,
                                      candidates | known_break)
 
@@ -189,7 +197,19 @@ def callback(self, clusters, prefix, backlog=None, known_break=None):
 
         return processed + self.callback(backlog, prefix)
 
-    def _break_for_parallelism(self, scope, candidates, timestamp):
+    def _break_for_parallelism(self, scope, dim, timestamp):
+        candidates = dim._defines
+
+        # Do not fission for data locality reasons if there's enough potential
+        # parallelism in the inner Dimensions
+        try:
+            ispace, = {e.ispace for e in scope.exprs[:timestamp]}
+            _, ispace1 = ispace.split(dim)
+            if len(ispace1.itdims) > self.FISSION_THRESHOLD:
+                return False
+        except ValueError:
+            pass
+
         # `test` will be True if there's at least one data-dependence that would
         # break parallelism
         test = False