The solar wind has a density of about 6 atoms per cubic centimeter at Earth orbit. If we say it moves at 500 km/s at Earth orbit that means that a naked surface will be hit by 30 billion antiprotons per second. That is about 9 watt of annihilation energy per square meter - three orders of magnitude below the energy in sunlight, but in the form of gamma rays. During solar eruptions this may increase by a factor of 100 or 1000 for a day or so.
Over 4.5 billion years it would annihilate 7.1 kg surface material per square meter - not enough to remove small asteroids and moons, but definitely reshape their surface chemistry and texture.
It would also tend to activate nuclei by changing them to isotopes with one less proton (or just blowing up nuclei), making surface materials in space radioactive. Lots of carbon 14 (from nitrogen 14) and tritium (from helium 4). Meteors would bring the more long-lived isotopes to the ground.
Even if Earth is somewhat protected by the magnetic field, auroras would involve annihilation reactions beside high energy collisions.
Overall, this would be a pretty radioactive environment. Not sure what the general radiation level would be, but about 54 tons of newly activated meteor surface material would be arriving each day.