I have both XE1 and Nex 6.
I did a comparison shoot out once. Different lenses but similar in performance. The lens on the Nex is a Samyang 24mm F1.4 which gives warmer colours. As you can see IQ is close in detail and resolution with a slight edge to XE1 due to no AA filter.
http://www.pbase.com/gregbradley/image/149391206 Fuji XE1
http://www.pbase.com/gregbradley/image/149391334 Nex 6
I thought both had excellent white balance. I have not experienced poor indoor white balance. Perhaps it is a setting, perhaps I have not used it much in that situation.
Nex has several advantages over XE1:
1. IQ is slightly less but its really not that much.
2. RAW converter of course is standard and so well supported.
3. Tiltable LCD screen (I really like that its more useful than you would think).
4. Nex is a bit smaller and more compact. Its a modern feel and well built but with cheapish buttons.
5. AF was faster but after the Fuji firmware upgrades its probably a tie.
6. Nex is way better for video with stereo sound, better tracking and face recognition AF. Like MUCH better than Fuji's which is crude and an afterthought.
7. Nex implementation of focus peaking for using manual lenses is way better than Fuji's. So using an older but excellent manual focus lens is a breeze and quite enjoyable. Contax G Zeiss lenses 2nd hand are cheaper than Fujis and of similar quality just manual focus. Minolta Rokkor lenses are also great value with fabulous Bokeh and a pleasing rendtion.
8. Nex 6 has wifi plus gps plus iphone remote control.
9. The Zeiss 24 F1.8 is a great lens and a lot of people love that one. Its a useful focal length and Fuji has deliberately held the 23mm F1.4 until late in the game to protect X100s sales.
10. Nex 6 has auto ISO a feature on the wish list for XE1/XPro 1 often complained about.
11. Nex 6 has the latest gen Exmor 16.3 sensor with phase detect AF that is also in the Fuji X100s. XE1 is the earlier version of that sensor.
12. EVF is slightly better on Nex.
13. You can buy and install apps on the Sony some of which are very good.
14. I am not 100% on this but I think most likely better battery life - could be wrong. I found XE1 battery life to be poor.
Pluses for XE1:
1. Overall Fuji lenses offer a better selection are better quality than Sony. Sony are cheaper on most but being on the Nex forum you hear a lot of complaints about the lack of high quality Sony lenses. The SL35 and 50 are well regarded, the 18-200 also (expensive), the Zeiss 24mm F1.8. Some like the Touits and they are expensive but fairly good but have negatives.
So as lenses account for a large percentage of IQ then Fuji has a large edge here at this time.
2. Styling is personal but most seem to prefer X system retro looks and external knobs. XE1 exposure compensation dial though is a liability in that it is easily knocked in the bag and will throw off a lot of your photos until you discover its moved off zero.
3. Slightly sharper images.
4. Better low light performance (quite a bit better but Nex is good, XE1 is superb and is the current king there).
5. Fuji jpegs out of the camera are famous for being so good. The colour in Fuji images is divine and people look great in Fuji images.
6. The menu system is way better. Sony's looks like the Managing Director's 8 year old kid designed it.
7. Both have sweep panorama function but Fuji's works better and is less fussy than Nex version.
So in the end the Nex has a lot of pluses but as you say IQ is everything and I chose to use XE1 as my primary camera because of that. The Fuji colours, white balance and image quality enhanced by the excellent range of superb lenses are reasonable cost is very seductive. Nex is a great camera and I eagerly await the full frame Nex so there you have it. XE2/XPro 2 unless they come up with a new sensor will in a way be merely catching up with Nex and the next Gen Nex will be a step ahead again. So Nex is further along/closer to the cutting edge of latest technology than Fuji.
Greg.