Why is this buffered rail splitter oscillating and how can I compensate it?
What is the reason for this oscillation with some capacitance between
the new ground and the symmetrical supply rails? I would have assumed
these caps would even increase the stability, as the feedback to U1 is
now somewhat low pass filtered.

Just because you have buffered the op-amp with an emitter follower push-pull circuit does not mean you have somehow isolated the op-amp from oscillating when you close the feedback loop. Pretty much any op-amp will sing its little heart out when the loading capacitance rises to a certain value and might only stop singing when the capacitance has risen to an unfeasibly and not useful large value of thousands of micro farads.

Can I compensate for that behaviour? How?

You can stop it by slugging the op-amp response with a local integration capacitor from op-amp output to inverting input but, make sure you use a 1 kΩ to 10 kΩ resistor to connect the inverting input to the emitter follower output.

My answer to this question contains a detailed breakdown of why an op-amp (buffered or not) will sing when capacitance rises to a certain small value and why it won't stop oscillating until the loading capacitance reaches a much, much higher value.