diff --git a/mlir/lib/Dialect/Linalg/Transforms/FusionOnTensors.cpp b/mlir/lib/Dialect/Linalg/Transforms/FusionOnTensors.cpp
--- a/mlir/lib/Dialect/Linalg/Transforms/FusionOnTensors.cpp
+++ b/mlir/lib/Dialect/Linalg/Transforms/FusionOnTensors.cpp
@@ -30,61 +30,26 @@
 // StructuredOp specific helpers.
 //===----------------------------------------------------------------------===//
 
-/// Relate the producer to the consumer loop iterations that access the same
-/// producer result element:
-///  consumerToProducerLoops =
-///   inverse(producerIndexingMap).compose(consumerIndexingMap).
-/// Return `consumerToProducerLoops` or none if the inversion fails.
-static Optional<AffineMap>
-getConsumerToProducerLoopsMap(AffineMap producerIndexingMap,
-                              AffineMap consumerIndexingMap) {
-  assert(consumerIndexingMap.getNumResults() ==
-             producerIndexingMap.getNumResults() &&
-         "expect the number of indexing map results to match");
-  // Ensure the producer indexing map is a projected permutation.
-  if (!producerIndexingMap.isProjectedPermutation())
-    return None;
-  AffineMap inverseIndexingMap =
-      inverseAndBroadcastProjectedPermuation(producerIndexingMap);
-  return inverseIndexingMap.compose(consumerIndexingMap);
-}
-
-/// Returns the producer result slice dimensions tiled by the tile loop nest or
-/// an empty vector if `getConsumerToProducerLoopsMap` returns none.
-// TODO: replace by Fourier-Motzkin and/or compute starting from consumer.
-SmallVector<int64_t> getTiledSliceDims(OpResult producerResult,
-                                       OpOperand *consumerOperand,
+/// Returns the tiled slice dimensions given the tiled consumer loop dimensions.
+/// The slice defines a hyper rectangular iteration space and fusing the
+/// producer is always possible. However, depending on the consumer indexing
+/// map, not all slice elements may be consumed and the tiles may overlap. In
+/// these cases, fusion introduces redundant computation.
+SmallVector<int64_t> getTiledSliceDims(OpOperand *consumerOperand,
                                        ArrayRef<int64_t> tiledLoopDims) {
+  // Get the consumer operand indexing map.
   LinalgOp consumerOp = consumerOperand->getOwner();
-  LinalgOp producerOp = producerResult.getOwner();
-  OpOperand *opOperand =
-      producerOp.getOutputOperand(producerResult.getResultNumber());
-
-  // Compute the `consumerToProducerLoopsMap` and exit if the computation fails.
-  AffineMap producerIndexingMap = producerOp.getTiedIndexingMap(opOperand);
-  Optional<AffineMap> consumerToProducerLoopsMap =
-      getConsumerToProducerLoopsMap(
-          producerIndexingMap, consumerOp.getTiedIndexingMap(consumerOperand));
-  if (!consumerToProducerLoopsMap.hasValue())
-    return {};
-
-  // Compute the set of tiled producer loops.
-  DenseSet<int64_t> tiledProducerLoops;
-  for (auto en : enumerate(consumerToProducerLoopsMap->getResults())) {
-    for (int64_t dim : tiledLoopDims) {
-      if (en.value().isFunctionOfDim(dim))
-        tiledProducerLoops.insert(en.index());
+  AffineMap indexingMap = consumerOp.getTiedIndexingMap(consumerOperand);
+
+  // Search the slice dimensions tiled by a tile loop dimension.
+  DenseSet<int64_t> tiledSliceDims;
+  for (auto en : enumerate(indexingMap.getResults())) {
+    for (auto tiledLoopDim : tiledLoopDims) {
+      if (en.value().isFunctionOfDim(tiledLoopDim))
+        tiledSliceDims.insert(en.index());
     }
   }
-
-  // Compute the slice dimensions for the tiled producer loops.
-  SmallVector<int64_t> tiledSliceDims;
-  for (auto en : enumerate(producerIndexingMap.getResults())) {
-    auto dimExpr = en.value().dyn_cast<AffineDimExpr>();
-    if (dimExpr && tiledProducerLoops.count(dimExpr.getPosition()) != 0)
-      tiledSliceDims.push_back(en.index());
-  }
-  return tiledSliceDims;
+  return {tiledSliceDims.begin(), tiledSliceDims.end()};
 }
 
 /// Returns the producer fused in place of `sliceOp`. Tile the producer operands
@@ -328,9 +293,10 @@
   if (!producerResult || !isa<LinalgOp>(producerResult.getOwner()))
     return failure();
 
-  // Compute the slice dimensions tiled by `tileLoopNest`.
+  // Compute the tiled producer slice dimensions given the tiled root operation
+  // loop dimensions `loopDims`.
   SmallVector<int64_t> tiledSliceDims =
-      getTiledSliceDims(producerResult, rootOpOperand, loopDims);
+      getTiledSliceDims(rootOpOperand, loopDims);
   if (tiledSliceDims.empty())
     return failure();
 
diff --git a/mlir/test/Dialect/Linalg/tile-and-fuse-on-tensors.mlir b/mlir/test/Dialect/Linalg/tile-and-fuse-on-tensors.mlir
--- a/mlir/test/Dialect/Linalg/tile-and-fuse-on-tensors.mlir
+++ b/mlir/test/Dialect/Linalg/tile-and-fuse-on-tensors.mlir
@@ -230,3 +230,39 @@
   return %1 : tensor<24x25xi32>
 }
 
+// -----
+
+//  CHECK-DAG:  #[[MAP0:.*]] = affine_map<(d0, d1) -> (d0 + d1)>
+//  CHECK-DAG:  #[[MAP1:.*]] = affine_map<(d0, d1) -> (8, -d0 - d1 + 18)>
+//  CHECK-DAG:  #[[MAP2:.*]] = affine_map<(d0, d1, d2) -> (d0, -d1 - d2 + 18)>
+#map0 = affine_map<(d0, d1) -> (d0, d0 + d1)>
+#map1 = affine_map<(d0, d1) -> (d0, d1)>
+
+//      CHECK:  fuse_non_rectangular
+// CHECK-SAME:    %[[ARG0:[0-9a-zA-Z]*]]: tensor<10x18xf32>
+func @fuse_non_rectangular(%arg0: tensor<10x18xf32>,
+                           %arg1: tensor<10x8xf32>) -> tensor<10x8xf32> {
+  %cst = constant 0.000000e+00 : f32
+  %0 = linalg.fill(%cst, %arg0) : f32, tensor<10x18xf32> -> tensor<10x18xf32>
+
+  //      CHECK:  scf.for %[[IV0:[0-9a-zA-Z]*]] =
+  //      CHECK:    scf.for %[[IV1:[0-9a-zA-Z]*]] =
+
+  // Compute producer on a hyper rectangular bounding box. Along the second dimenson,
+  // the offset is set to the sum of the induction variables and the upper bound
+  // to either eight (sum of the tile sizes) or eighteen (sum of the domain sizes)
+  // minus the induction variables.
+  //      CHECK:      %[[SUM:.*]] = affine.apply #[[MAP0]](%[[IV1]], %[[IV0]]
+  //      CHECK:      %[[TS1:.*]] = affine.min #[[MAP1]](%[[IV1]], %[[IV0]]
+  //      CHECK:      %[[UB1:.*]] = affine.min #[[MAP2]](%[[TS1]], %[[IV1]], %[[IV0]]
+  //      CHECK:      %[[T0:.*]] = tensor.extract_slice %[[ARG0]]
+  // CHECK-SAME:                                        %[[IV1]], %[[SUM]]
+  // CHECK-SAME:                                                , %[[UB1]]
+  //      CHECK:      %[[T1:.*]] = linalg.fill(%{{.*}}, %[[T0]])
+  %1 = linalg.generic {indexing_maps = [#map0, #map1], iterator_types = ["parallel", "parallel"]} ins(%0 : tensor<10x18xf32>) outs(%arg1 : tensor<10x8xf32>) {
+  ^bb0(%arg2: f32, %arg3: f32):  // no predecessors
+    %2 = addf %arg2, %arg3 : f32
+    linalg.yield %2 : f32
+  } -> tensor<10x8xf32>
+  return %1 : tensor<10x8xf32>
+}