Electrical question regarding CF bulb use

We have a fixture that consists of 4 standard bulb sockets, each socket rated up to 60 Watts - no dimmer. I have no desire to burn 240W while eating, so I switched the bulbs to compact fluorescent (much like we have in the rest of the house).

Now here’s the odd thing. If put in 4 CF bulbs, they will all flicker, enough to drive you mad, but if I replace any one of the 4 with a standard incandescent bulb, the flickering stops. (I’ve rotated the CF bulbs; I’ve swapped out each one individually with an incandescent one and get the same behavior).

Any idea what could be happening here?

For clarity, here is a list of all combos tried:
CF-CF-CF-CF = flicker
I-CF-CF-CF = no flicker
CF-I-CF-CF = no flicker
CF-CF-I-CF = no flicker
CF-CF-CF-I = no flicker

Forgot to mention that I also had swapped out each individual CF (from the original 4) with a fifth bulb to make sure I didn’t have a bum bulb; but I guess I could have had multiple bad CFs, but I didn’t think that likely.

My WAG is that there’s some kind of impedance-based smoothing going on in the bulb filament, or something like that.

One thing they all have in common is the phase of the power being applied. That’s very likely to be link to the observed flicker.

4 CFs is probably reaching some threshold of mutual effect (e.g. transient current draws from the instant of ionization damping ionization in other CFs in the fixture; enough UV leaking past the phosphor of three neighbors to influence each bulb in close proximity; etc.)

To test this general mechanism, try using only 3 CFs (maybe rotating the empty socket).