Amplify a 10mVpp signal 100 times and get a clean output waveform.

TL072 pinout

Ga $G = -{R_2 \over R_1} = -{100000 \over 1000} = -1000 $

Output signal is looking good. Not entirely clean (some wiggles at the peaks and valleys), but with the cable mess and on a breadboard this might be the best I can hope for.

There is a noticeable upwards DC shift, what might be the cause? Reproducing the circuit in Falstad does not do that:

Does the same happen when I use a different op-amp?

Interestingly, the LM358 also has a DC shift but in the other direction:

I read about an input offset voltage of op-amps here and wonder how high it is for the IC's I am using.
For the LM358 it is typically 2mV. Multiplied by a gain of 100, no wonder there is such a shift:

Even more for the TL072:

How do I mitigate this? In the StackExchange article mentioned above people say:
Even better, Dave from EEVBlog made a video about it! In the comments below the video I read these ideas:
"Manufacturers recommend to balance opamp inputs against bias current...need to ground (+) input via 0.99kOhm resistor" (no effect)
"With a chopper amp, there is internal switching, correct way to decouple would be across the power pins, not to ground, as latter could inject noise"
"Inputs aren't seeing the same impedance... connect the non-inverting input to ground via a 1k resistor"
"using just a bypass cap between V+ and V- (instead of two to ground) usually reduces noise"
Also, in the forum thread about the video:
I also tried a 0.1uF cap on both Vcc+ and Vcc- to gnd but it does not make a difference. Perhaps a pot is the only option?
Yes. What did work for me was this little add-on circuit I found online:

NOTE: The 100k resistor on IN+ is important for when the potmeter is turned all the way to one side.
That way the potentiometer can be carefully balanced to generate an output around averaging 0V. This is the result:
