The keyword to search for, if you are interested, is epigenetics. Modification to e.g. histone protiens can change the expression of genes through a variety of mechanisms. As far as I'm aware, exactly how this works isn't that well understood.

Just a note: in this case, histones are irrelevant, since prokaryotes don't have any. But more generally, protein binding, modification of bound proteins, and methylation of the DNA itself are all epigenetic mechanisms that could be in play here.

Prokaryotes actually do have histones! Or at least histone-like proteins. https://link.springer.com/article/10.1134/S0003683811060020

To add to the list: activity of restriction enzymes, especially specific ones.

Colistin has a pretty interesting mechanism of action, disrupting ionic balance in cell wall of lipopolysaccharide.

The mechanism of resistance may well be lucky upregulation of ionic balance and cell wall stabilization proteins in response to the attack. Same as for lysozyme.

This kind of ionic environment has wide ranging effects on cell metabolism so changes genre expression. In multiple generations, the resistance might get baked into the genome...

The OP says that the genome sequence hasn't changed. So in order to have that persistent upregulation without any change in the genome sequence, you (most likely) need an epigenetic mechanism to explain it.

> histones are irrelevant, since prokaryotes don't have any

Good point :).

Genes are expressed in response to a signal, and no two bacteria in a population are ever receiving exactly the same signal, hence heterogeneity of gene expression.

Some bacteria have a "persistent" phenotype which means that they slow their metabolism way down, stop replicating and just hang out although they are genetically identical to the rest of the population. Normally (or in nutrient-rich conditions) this puts them at a disadvantage vs. the rest of the population, but in the event that there is an antibiotic compound the persisters survive way better. Persisters make up a very small percentage of the total population but can end up re-seeding the entire population after antibiotic treatment. It's not clear how the persister phenotype is regulated, so scientists often refer to it as stochastic. The really interesting thing is that persisters and non-persisters are genetically identical, and it's thought that maintaining some level of persisters in population is a bet-hedging strategy of the population.

*will edit with sources in a little bit

The simple answer is that no two individuals are subject to identical stimuli. The hard work is understanding what the relevant differences in stimuli are.