I see several related questions, mostly taking the standard position/interpretation that complex-valued amplitudes aren't "physically real" (though their imaginary components do participate in interference effects that are ultimately experimentally observable), whereas real-valued probabilities are "physically real". And I'd agree with that.
However, although probabilities are observable, they aren't canonical operators in the usual Hilbert space or $C^\ast$-algebra sense. You "observe" probabilities by repetitions of some relevant measurement, and then applying the standard Kolmogorov frequentist interpretation of probability to the set of resulting outcomes.
So that's hardly a Hermitian operator, or anything axiomatically identified with typical "observables". It's pretty much just the Born rule. So if we'd agree that bona fide observables are "physically real", then calling probabilities "physically real" must be using that term in some different sense. So what are these two distinct senses of "physically real"; in particular, what distinguishes the "physical reality" of probabilities from that of typical observables?
Edit Note that the Born-rule probability calculation is foundationally different than, say, the calculation of an apple's volume from a measurement of its radius, which we might call the sphere-rule. You can design an experimental apparatus that directly and independently measures an apple's volume (though I'd refrain from running naked through the streets crying "Eureka!" about it:), whereas there's no possible way for an apparatus to >>directly<< measure probabilities at one fell swoop, so to speak.
I mention this in response to Ion-Sme's answer: Since there are direct measurements of gravitational fields, that wouldn't be an apt analogy relevant to the purpose at hand.
Edit The comments from @Nathaniel and answers from @BobBee and @StéphaneRollandin seem to all conclude (in Stéphane's words) that it "depends on what you mean by 'real'". That kind of seems to suggest there's no definitive answer, and even the definition-dependent "answers" here are basically discussions, i.e., nothing formal and rigorous. So let me try (without completely succeeding) to make the question a little more definite, as follows...
In the current (March 2018) issue of APS News, there's a page 4 obit for Polchinski, where they quote from his 2017 arXiv memoir, in part, "I have not achieved ... explaining why there is something rather than nothing." So can we agree with his conclusion that there's something rather than nothing? Okay, then in that case, I'd suggest that those non-nothing somethings are, by our definition here, "real".
I realize that's still pretty wishy-washy, almost certainly insufficient for any subsequent rigorous treatment stemming from those words. And that's kind of the whole point. We've got Polchinski saying, "there's something rather than nothing", and he presumably wasn't wishy-washy philosophizing. Instead, he was lamenting over not achieving that ambition of explaining why there's this "something". But if he'd wanted to rigorously mathematically achieve that explanation, it would prerequisitely require some rigorous axiomatic-worthy definition of the "something" you're going to subsequently explain.
So this doesn't really define "something"~"real", but I think it's a reasonable argument why such a formal definition is necessary, and physics-worthy rather than just philosophy-worthy. Physics formally, rigorously, mathematically discusses the observable "somethings" of experience. But then it can't say what those "somethings" are??? Seems like that's an important missing piece of the puzzle.