CMV: COLA constraint is not necessary
You just triggered one  old itch, so this is a partial (both meanings) answer only. For discrete-time, uniformly sampled STFT, one often uses the usual fixed-window with $3/4$, $1/2$, $1/4$ overlaps, and somehow uses  direct closed form inverses. However, as long as there is redundancy (and completeness), there exists a infinite option for inverses.

In a   collaborative work (with Jean-Christophe Pesquet and Jérôme Gauthier), following other works on vector-frames, we used the formalism of complex oversampled multirate filter banks 

to model analysis STFT, with the aims of


checking conditions under which the STFT were invertible, given a hop between time-frames,
building "better" inverses (better localized in time or frequency) with optimization.


Our aim at that time was double:


for data in higher dimensions $d$, reduce the STFT redundancy $r^d$ as far as we could,
-for 1D data, reduce the number of channels and increase the hop as we could as similar quality for real-time event detection.


All was based on polyphase matices, as we did consider windowed transform only as a special case;  so the main outcomes were the following:


given any redundant analysis filter bank (complex or not, windowed-DFT like or not), check whether it is invertible
in most cases, anything redundant is almost surely inversible,
windows with negative values can be used, liked Kaiser windows, see On the Optimization of Oversampled DFT Filter Banks, 2007

you can reduce the degrees of freedom in the oversampled synthesis to perform better concentrated inverses after selection/denoising for (close-to) real-time computations,
ensure the Hermitian character in the inverse for scalar threshold.


There were remaining issues, among which: the difficulty to go back to the orthogonal setting when the redundancy tends to one. Weirds behaviors were observed, possibly related to Farey sequences, but I have no definitive clue. And 
References can be found in Optimization of Synthesis Oversampled Complex Filter Banks, 2009. Some related codes is available at SURE-LET Optimal oversampled Complex Filter Banks synthesis toolbox. We do plan a better release, but it has been remaining upcoming for months.

However, we did not investigate the non-COLA effect, and the necessary condition is not stated. My belief is that it is not necessary (due to redundancy), but this should be proven.