You cannot avoid the problem with space dust if the planet is similar to earth. A nice article on cosmic dust shows a number of aspects of the matter hitting our plant daily. The total mass is expected to be somewhere in the range of 5-300 tons daily - yes the number is not very precise, but at least it gives us the general size of the problem. Even at the minimum 5 tons per day, that is a lot of matter.
The article states: Particles with diameters greater than about 2 millimeters produce visible “shooting stars,” but most of the mass of dust particles entering the atmosphere is estimated to be much smaller than this, so can be detected only using specialized meteor radars.
For a 2 mm sphere at an average density of 2.5 g/cm**3 the average mass per particle would be about 0.01 grams and the planet is being bombarded by something like 500,000,000 of these each day. Likely quite a bit more particles as most of the dust is quite a bit smaller than 2mm. Just how perfect do you think your planetary defense system is given the incoming velocities of about 38-248 thousand km/hr and diameter mostly less than 2 mm. 50,000 impacts per day would occur for a 99.99% effective defense.
At 0.01 grams, the incoming particles will have a yields of a 0.5 kt bomb, with a very characteristic radiation signature. Most will be much smaller, but still very easily detected. Furthermore, since most of the cosmic dust will be in the ecliptic plane, there will be a pattern that these events are more common over the equator than the poles (due to the changing angle of incidence). Likewise there will be patterns of heavier activity that correspond to meteor shows. Because of the earth's orbital velocity (30 km/s) around the sun, the number of impacts at the leading edge will be significantly higher than the trailing edge and this will definitely be noticed i.e., the rate at dawn will be higher than at dusk.
The jig will be up long before spaceflight if this anti-earth had the same development history as us. In fact the bright flashes in the sky will make this a very well studied and obvious problem long before spaceflight, though without an understanding of particle physics will be inexplicable. In fact I believe that the light shows would drive an interest in astronomy.
The orbital defense grid needed to be 99.99% effective will be directly observable from the planets surface. You would need large arrays of Hubble-sized telescopes or smaller numbers of more massive telescope facilities for detection. Then you need the defensive equipment and power sources for the actual defense. These will be blatantly obvious to 1950's tech ground based telescopes.
As mentioned by bowlturner, the earth also loses mass -- about 250 tons per day (estimates vary), mostly hydrogen, a little helium and other traces. Some of this will result in noticeable effects from the planets surface too. However, since the bulk of this will be in the form of single atoms and molecules the effect will be much less noticeable than the daily light shows. Once noticed however, there will be an obvious correlation that the effect is a result of something coming from the planet.
I remembered learning about putting micro-meteoroid detectors in early spacecraft in school and found this article on the first one launched. On the Explorer 1 mission launched Feb 1 1958 detected 145 impacts during 11 days for an effective detector area of 0.075 sq. meters. This is 175 impacts per day per sq meter. The detector was designed to detect impacts of 0.01 mm diameter. The surface of the earth is about 510 billion sq. m, so the number of impacts per day is about 90 trillion per day. At a 0.01 mm level, the antimatter explosion would be about the same as a stick of dynamite so I expect quite a few microflashes would be visible in any night sky. An even worse situation for this particular scenario. There are good reasons that astronomers are fairly certain that no large scale antimatter objects exists in the universe.
For those of you that think a 0.5 kton explosion of TNT is not that large, watch the video of Operation Sail Hat.
I forgot about solar wind - but after doing the math, I expect less about 0.28 kg worth of solar wind striking the earth every second - ignoring the effect of earths magnetosphere, but what of course, the solar wind does not ignore the magnetosphere - I would expect the auroras to be supercharged.
I did overlook one other thing when I originally answered the question. As bulk matter the dominant annihilation would be that of proton/anti-proton or neutron/anti-neutron pairs. This type of radiation is not the gamma-ray characteristically produced by electron/positron annihilation that we tend to think of (511 keV). So, the radiation would be even higher energy levels than 511 keV, but the quark annihilations result very short-lived pions that further decay - but this decay also results in neutrinos which carry away over 20% of the original mass-energy meaning that the local explosive effect is somewhat less than expected from 100% mass-energy conversion.
With current satellite technology we do measure cosmic radiation resulting from positrons, anti-hydrogen, and even anti-helium. We do not detect cosmic rays that suggest there is anti-matter source equivalent to a anti-matter mirror-image source of material. I.e, the cosmic rays indicate that the anti-matter striking us is generated by high-energy physics style anti-matter generation, not the result of anti-stars exploding.
I found a New Scientist article announcing the first production of anti-helium in 2011. That should indicate how hard it is to produce heavier anti-elements via high-energy physics. The article also mentions the anti-lithium is "beyond the reach of today’s colliders" Even the theoretical Big Bang nucleosynthesis produces only small traces of any element heavier than Lithium (1 part in 1E15), Lithium itself being only 1 part in 1E9.
Making your own anti-matter planet will be a bit of a challenge too - a very advanced civilization that constructed an earth-like planet out of anti-matter would have really gone to a lot of effort as it appears that you have to make your own anti-matter from scratch.
