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Particles disappear when they meet their anti-particle.

So don't.

Select "non-matching" matter and anti-matter particles.

Could you combine any combination of particles including at least 1 matter and 1 anti-matter, to create a stable system like an atom?

Why/why not?

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Simply put, there are no stable charged particles beyond electrons, protons, and their antiparticles. This means that if you want a pair of oppositely charged particles, it needs to be $p^+e^-$ (hydrogen), $p^-e^+$ (antihydrogen), $e^+e^-$ (positronium), or $p^+p^-$ (protonium).

If you wish to use any "non-matching" particles, then at least one of the components must be unstable. The easiest example of this is muonium, $p^+\mu^-$, made from a proton and a muon. While the two won't annihilate, the muon will decay into an electron and a pair of neutrinos within a few microseconds. Moreover, that's about the longest that you can get any such scheme to work.

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  • $\begingroup$ Well, you can imagine trying to form an atom of quasi-deuterium of a proton, electron and an anti-neutron. But it won't work: the proton and anti-neutron will annihilate at the valence quark level resulting in a spray of hadronic junk. $\endgroup$ – dmckee --- ex-moderator kitten Nov 10 '16 at 17:32
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An atom is (typically) stable not because its composed of matter or antimatter as opposed to a mixture of matter and antimatter, but because the particular particles involved are stable by themselves and because the probability of the electron being in the nucleus is (typically) zero. So in theory you certainly could create an atom-like structure of matter and anti-matter using dissimilar particles, but as Emilio Pisanty says, we don't know of any such sets of particles.

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