TIFF images with malformed EXIF data are refused by Django with Pillow 3.0.0

A few TIFF images are refused by Django in ImageField when using Pillow 3.0.0. When using Pillow 2.7.0, the images are accepted, but with a warning suggesting that some EXIF metadata is incorrect:

/home/zorun/tmp/venv2-celutz/lib/python2.7/site-packages/PIL/TiffImagePlugin.py:466: UserWarning: Possibly corrupt EXIF data.  Expecting to read 616 bytes but only got 76. Skipping tag 279
  "Skipping tag %s" % (size, len(data), tag))

/home/zorun/tmp/venv2-celutz/lib/python2.7/site-packages/PIL/TiffImagePlugin.py:466: UserWarning: Possibly corrupt EXIF data.  Expecting to read 8 bytes but only got 0. Skipping tag 282
  "Skipping tag %s" % (size, len(data), tag))

/home/zorun/tmp/venv2-celutz/lib/python2.7/site-packages/PIL/TiffImagePlugin.py:466: UserWarning: Possibly corrupt EXIF data.  Expecting to read 8 bytes but only got 0. Skipping tag 283
  "Skipping tag %s" % (size, len(data), tag))

/home/zorun/tmp/venv2-celutz/lib/python2.7/site-packages/PIL/TiffImagePlugin.py:466: UserWarning: Possibly corrupt EXIF data.  Expecting to read 8 bytes but only got 0. Skipping tag 286
  "Skipping tag %s" % (size, len(data), tag))

/home/zorun/tmp/venv2-celutz/lib/python2.7/site-packages/PIL/TiffImagePlugin.py:466: UserWarning: Possibly corrupt EXIF data.  Expecting to read 8 bytes but only got 0. Skipping tag 287
  "Skipping tag %s" % (size, len(data), tag))

/home/zorun/tmp/venv2-celutz/lib/python2.7/site-packages/PIL/TiffImagePlugin.py:466: UserWarning: Possibly corrupt EXIF data.  Expecting to read 28 bytes but only got 0. Skipping tag 305
  "Skipping tag %s" % (size, len(data), tag))

/home/zorun/tmp/venv2-celutz/lib/python2.7/site-packages/PIL/TiffImagePlugin.py:466: UserWarning: Possibly corrupt EXIF data.  Expecting to read 20 bytes but only got 0. Skipping tag 306
  "Skipping tag %s" % (size, len(data), tag))

/home/zorun/tmp/venv2-celutz/lib/python2.7/site-packages/PIL/TiffImagePlugin.py:466: UserWarning: Possibly corrupt EXIF data.  Expecting to read 8 bytes but only got 0. Skipping tag 339
  "Skipping tag %s" % (size, len(data), tag))

With Pillow 3.0.0, tile generation also fails on those TIFF files:

[2015-10-23 18:27:11,478: ERROR/MainProcess] Task panorama.tasks.generate_tiles[487453ea-25cf-488a-9059-1eec75066889] raised unexpected: OSError("cannot identify image file '/home/zorun/illyse/git/celutz/media/pano/toit.tif'",)
Traceback (most recent call last):
  File "/home/zorun/tmp/venv3-celutz/lib/python3.5/site-packages/celery/app/trace.py", line 240, in trace_task
    R = retval = fun(*args, **kwargs)
  File "/home/zorun/tmp/venv3-celutz/lib/python3.5/site-packages/celery/app/trace.py", line 438, in __protected_call__
    return self.run(*args, **kwargs)
  File "/home/zorun/illyse/git/celutz/panorama/tasks.py", line 11, in generate_tiles
    return gen_tiles(*args, **kwargs)
  File "/home/zorun/illyse/git/celutz/panorama/gen_tiles.py", line 23, in gen_tiles
    orig_im = PIL.Image.open(image)
  File "/home/zorun/tmp/venv3-celutz/lib/python3.5/site-packages/PIL/Image.py", line 2286, in open
    % (filename if filename else fp))
OSError: cannot identify image file '/home/zorun/illyse/git/celutz/media/pano/toit.tif'