That's a tough one. I even suspect that any rigorous definition of a physical object would contradict what we know of physics, hence proving that the idea of a totally "physical" object is naive. For example, if physicality is anything, it's mechanistic. Yet how can an object be 100% mechanistic when it accords with the principles of quantum mechanics which includes behavior that's fundamentally unpredictable? I mean you might say that if we had enough information and knew all the rules we could predict even quantum events, but that's purely speculative. Also any such model would have to be non-local (per Bell's inequality), and aren't objects pretty much localized by definition?

As another example, we could say that a physical object can't be something abstract like a wave (a wave is physical, but it's not an object), yet in QM we know that every object corresponds to a wave, the bigger the object the higher the frequency of the wave.

I could say that for an object to be physical it must (aside from being an object) be detectable by physical instruments. But there are two problems with that. (1) is that it's circular: how do you know if a detecting instrument is physical or not? And (2) is that it's all one reality, and if anything non-physical exists in reality then it must affect physical things in one way or another or we could never know of (and hence speak of) them. So that implies that non-physical things (like a soul) may be detectable by physical instruments if only we know what to look for.

To solve (1), I could say that a physical object must be detectable by instruments that we can read with our physical senses, where a "physical sense" is any sense that's not principally psychic, such as sight, smell, taste, hearing, and touch (mainly sight and sound). But that leaves problem 2.

To solve (2) I could say that every aspect of the thing in question, down to its essence, must be detectable by the physical instrument. The problem with that, though, if we have no idea, and no way of knowing, if regular everyday physical things are physically detectable in every aspect. If we could mathematically model something and predict 100% of its behavior with the appropriate inputs, then we could surmise that we've accounted for 100% of the object. But again, per QM we can never 100% predict anything's behavior. Therefore we don't know what "more to the story" there may be behind "physical" objects at large.

Another tempting way to solve (2) would be to say that the object must be directly detectable by physical instruments as opposed to indirectly. but the problem with that is that there are plenty of things we think of as physical that we can only indirectly detect, such as, for example, quarks.

So, again, my strong suspicion is that nothing is really physical in the way we think of "physical"--there's always more to the story, way more. However, given that we do have a common-sense idea of what things are (or at least appear) "physical" vs. what things aren't, it may be possible to come up with a definition of "physical" that satisfies that common-sense idea, at least for the most part. But I think that definition would be rather weak and ad hoc.