Is it possible that antimatter has positive inertial mass but negative gravitational mass?

A long comment:

AEGIS is a collaboration of physicists from all over Europe. In the first phase of the experiment, the AEGIS team is using antiprotons from the Antiproton Decelerator to make a beam of antihydrogen atoms. They then pass the antihydrogen beam through an instrument called a Moire deflectometer coupled to a position-sensitive detector to measure the strength of the gravitational interaction between matter and antimatter to a precision of 1%.

A system of gratings in the deflectometer splits the antihydrogen beam into parallel rays, forming a periodic pattern. From this pattern, the physicists can measure how much the antihydrogen beam drops during its horizontal flight. Combining this shift with the time each atom takes to fly and fall, the AEGIS team can then determine the strength of the gravitational force between the Earth and the antihydrogen atoms.

Also new experiments are in the process. In total there are three experiments at CERN to measure the effect of the earth's gravitational field on antimatter. Patience.

I suppose nothing is impossible, including the sign of gravity for ordinary matter reversing direction tomorrow, but it seems really extraordinarily unlikely that antimatter falls up.

It is absolutely precluded by general relativity (more specifically the equivalence principle) that different particles gravitate in different ways. For antihydrogen to fall up would require either that general relativity is wrong or that there is a new, hitherto undetected long-range force countering gravity. And GR would have to be hugely wrong, and the new long-range force would have to be quite strong, to make antihydrogen actually fall up rather than merely falling down at a slightly different acceleration. It's just not plausible that we would have failed to see a fundamental effect of that size for all this time.

Another reason to doubt that antimatter has weird properties is that "antiness" isn't actually an attribute of particles in quantum field theory. Protons and antiprotons are antiparticles of each other; neither one is the antiparticle. Antiprotons get the "anti" prefix merely because they're less common. Some particles (photons, for example) are antiparticles of themselves. We know that photons fall down (from the bending of starlight by the sun, for example). If antiparticles antigravitate then photons would have to fall up also, which isn't even self consistent. I wrote about this in more detail in another answer from which I copied some of the above text.

Other answers mention the AEGIS and ALPHA experiments, but note that AEGIS is looking for deviations in the acceleration "to a precision of 1%", and ALPHA seems to be going for similar precision. They aren't expecting antihydrogen to fall up; testing for that would require only a precision of... 200%, I suppose. No one, to my knowledge, expects antihydrogen to fall up.

You are asking whether antiparticles have different gravitational mass than particles.

There is a very good example of why the answer to your question is no, and that is light itself.

Whether you treat light classically or quantum mechanically, in both cases you will see that light or their quanta, photons do bend spacetime, they have their own gravitational effects and they do bend spacetime.

Given that photons have energy and momentum, it would surprise me if they do not induce curvature. I also note that the expansion of the "radiation-dominated" early universe was caused by what is generally described as a photon gas and not as a classical electromagnetic field. So the idea that photons bend spacetime is part of mainstream cosmology, such as the standard Lambda-CDM model.

Do photons bend spacetime or not?

Since photons have stress-energy, they do bend spacetime. And since they are their own antiparticles, the answer to your question is that particles and antiparticles have the same gravitational mass.