Some companies offer compact reusable bags that can be stored in a coat pocket. For example:

https://seatosummit.com/products/ultra-sil-day-pack - I've had a few different versions of this bag for years. There are also cheaper/bigger/different versions of the same sort of thing you can find online for "packable daypack".

https://nanobag.com/products/nanobag - I have heard good things about these, but I prefer a backpack because it allows me to be hands-free, or to use my hands to hold more items.

However, I would start by carrying a lightweight "single-use" plastic bag, and simply re-use it. Plastic bags are not as strong as these premium bags, but they hold up well enough to be useful in most scenarios.