operational amplifier - Precise Full-wave rectification

I've been for quite some time now, trying to create a precise full-wave rectification of a signal coming from a CT sensor SCT-013-030.

Here's the circuit:

^{simulate this circuit – Schematic created using CircuitLab}

Now, actually the 7805 voltage regulator is also providing regulated power supply to the remaining of the circuit, which is basically a XBee that's activating a given relay. If you feel it's required I'll update the post.

I first started by doing only a half-wave rectification. So I hadn't even the second op-amp in place, and the following was being displayed in the scope:

Basically the yellow waveform is Vin, blue waveform is V1. I must say that, considering the resistors (the same 10K at 1%), I can't actually understand that voltage boost.

When I try to apply the Full-wave rectification I get this:

Here's a zoomed screenshot of the last scope image:

I would probably say that I must first understand why am I getting such a boosted voltage on the output, while doing only a half-wave rectification.

The input signal is around 50hz. Another thing is that the bridge rectifier is actually a 4 diodes set up.

---

So, after compiling all the great comments and answers I tried to implement all the suggestions. And I must say that I'm rather confused.

Here's the current circuit. Note that I left out the AC/DC conversion to simplify:

^{simulate this circuit}

As a first approach I decided to do the full rectification, but without the dual power supply. For visibility purposes I've increased the power being used on the power cord being measured:

Afterward I've implemented the dual power supply making use of a inverting regulator ICL7660CPAZ, I got the following:

And last by not least I've disconnected whatever was attached to the power cord, and got this funky wave form:

I'm completely lost here...

---

So, I have to conclude that the problem is with my implementation of the precise full-wave rectifier. Even if the schematics that I'm presenting here, make sense, I'm problably not implementing it correctly.

So, bare it with me... but I feel compelled to post my breadboard image, because sincerely I'm out of ideas... the next step is to buy a different op-amp...

Let me know if you can see anything terribly wrong... needless to say that I've rebuilt the same circuit 3 different times...