I believe Entropy to be bang on. I have the Tamron 70-300 on Pentax mount. It does suffer from real bad purple fringing. It is super easy to eliminate it in your photos if you have Photoshop though. I only do this PP on the keepers.
Here is a quick and easy way to do this.
1. Use the eyedropper tool and select the background colour you want to replace the purple fringe with.
2. Select your paintbrush tool; choose a nice brush with a feathered edge.
3. In the paintbrush properties, select "colour". This will keep the luminance levels in your background photograph.
4. Using short strokes, paint the purple out of the photo.
It is that easy...or you could use a CA removal plug-in...but I have had not much luck with those
Here is an example of what you might expect with the Tamron 70-300. Honestly, the lens isn't that bad for the price.
And here is the final product with CA removal as per above method.
As for the Sigma. I have read a few online reviews stating that the APO designation didn’t seem to make much of a difference between the cheaper lens and the more expensive one when looking at quality of output. Just something to keep in mind.