Short answer, if you are asking about a counter Earth, it is impossible for a counter Earth to have remained undetected until now, but it is not totally, absolutely, 100 percent, impossible for a counter Earth to suddenly appear in our solar system, and it is much more possible for aliens to establish some sort of space base in the position of a hypothetical counter Earth.
The long answer begins.
The equatorial diameter of the Sun is about 1,392,784 kilometers
At the varying distances between the Earth and the Sun, the Sun has an angular diameter of 31 minutes and 27 seconds to 32 minutes and 32 seconds. So a cone of space between a human eye on Earth to the sun and beyond to Earth's orbit on the far side of the Sun would be a very thin cone expanding from zero at the eye to 1,392,784 kilometers at the distance of the Sun and to about 2,785,568 kilometers at the opposite side of the orbit of the Earth.
With Earth's elliptical orbit the distance between the Earth and the Sun varies from about 147,095,000 kilometers to about 152,100,000 kilometers. The average distance is about 149,598,023 kilometers. If Earth's orbit was perfectly circular with a radius of 149,598,023 kilometers Earth's orbit would have a circumference of about 939,951,280 kilometers. So one arc degree of Earth's orbit covers about 2,610,975.7 kilometers.
A space probe would have to travel about 1,392,784 kilometers ahead or behind Earth's position in Earth's orbit to a position where an object on the far side of the sun at Earth's orbital distance would appear above the edge of the Sun.
If, for example, a space probe is given 1 kilometer per second more or less than Earth's orbital velocity, it will move 1 kilometer ahead of or behind the Earth in every second, 60 kilometers in every minute, 3,600 kilometers in every hour, and 86,400 kilometers in every day. It will take 16.120 days to move 1,392,784 kilometers ahead of or behind the Earth, 32,240 days to move twice that distance, etc.
The Sun's corona, extending thousands and millions of kilometers above its surface, is very bright, but the million times brighter glare of the Sun makes it impossible for the human eye to notice the corona. Except during total solar eclipses when the Moon blocks the light from the Sun's surface, enabling humans on Earth to see the corona.
Bernard Lyot introduced the coronagraph in 1931 to block out the light from the Sun's surface and thus view the corona during the day.
Coronagraphs in outer space are much more effective than the same instruments would be if located on the ground. This is because the complete absence of atmospheric scattering eliminates the largest source of glare present in a terrestrial coronagraph. Several space missions such as NASA-ESA's SOHO, and NASA's SPARTAN, Solar Maximum Mission, and Skylab have used coronagraphs to study the outer reaches of the solar corona. The Hubble Space Telescope (HST) is able to perform coronagraphy using the Near Infrared Camera and Multi-Object Spectrometer (NICMOS),[5] and there are plans to have this capability on the James Webb Space Telescope (JWST) using its Near Infrared Camera (NIRCam) and Mid Infrared Instrument (MIRI).
https://en.wikipedia.org/wiki/Coronagraph [1]
so there have been a number of space based coronagraphs studying the Sun's corona, and there will be other's in the future.
Einstein's Theory of General Relativity predicted that gravity would bend light twice as much as in Newtonian physics. During a total eclipse of the Sun, stars can be seen near the Sun, and by measuring how far their images were from their true positions in the total eclipse of May 29, 1919 Dyson and Eddington, and Crommelin and Davidson, were able to confirm the predictions of General Ralativity.
So a space probe with a coronagraph should be able to make its own eclipse and see a counter Earth as soon as that space probe was far enough from Earth that the counter Earth was no longer behind the Sun from the position of the space probe, which would take a few days, weeks, or months depending on the speed and trajectory of the space probe relative to Earth.
And of course once a space probe got far enough from Earth that it could aim its telescope at the rough area where the counter Earth should be orbiting without including the Sun in the field of view, then it wouldn't need a coronagraph to take images of the area where the counter Earth should be.
So if there was a counter Earth orbiting exactly opposite to Earth, it would have been detected long ago by some space probe, and people at the space exploration stack exchange could probably tell you the latest possible date it could have remained undetected. They can probably link you to photos that would have shown the counter Earth if it was there.
https://www.google.com/search?q=space+exploration+stack+exchange&oq=space+exploration+stack&aqs=chrome.0.0j69i57j69i60l2j0j69i64.10407j0j7&sourceid=chrome&ie=UTF-8 2
But suppose that it is possible to create artificial wormholes. Perhaps an alien planet is under attack by their enemies. Perhaps the enemies use artificial wormholes to make planets appear with trajectories that will smash them into the besieged planet. So the natives of the planet under attack generate an artificial wormhole with a mouth right ahead of their planet in its orbit.
The planet disappears into the mouth of the wormhole, which disappears. The other mouth of the wormhole appears in interstellar space far away from the realm of their enemy - maybe hundreds, thousands, millions or billions of light years away. The planet exits the far wormhole mouth, which disappears.
The aliens can keep their planet lit and heated artificially, but desire to find a star to orbit for sentimental reasons maybe.
They will want to select a star with the same mass, luminosity, and age as their home stars, so that the radiation from that star will be exactly the same as the lifeforms on their planet are adapted to an so the orbital velocity of the planet, which it still retains will be exactly right.
So they study a bunch of nearby stars, and find one that seems perfect. Unfortunately, there is already a planet in the perfect orbit around that star. Fortunately, they could put their own planet into an exact opposite orbit around the star. Unfortunately, there are intelligent natives on the planet of that star. Fortunately, the natives are merely at the 21st century level of development, thousands of years behind, and putting their own planet in a counter obit will prevent the natives from finding out for a while, giving more time to prepare for any possible hostility.
So the aliens decide to use on or more artificial wormholes to put their planet into a counter Earth orbit on the far side of the Sun from Earth.
Or maybe some hypothetical aliens build a space bases on the far side of the Moon, so they can observe Earth without their base being seen by even the most advanced Earth based telescopes. For decades occasional science fiction stories pointed out that an alien base on the far side of the moon couldn't be detected from Earth.
And then the aliens discover that Earth is beginning the Space Age, and say some alien version of &(%^*%$${0%@^ !! and start building a space habitat on the far side of the sun, orbiting in the counter Earth position, and using its propulsion system from time to time to keep within the counter Earth position. They dismantle the base on the lunar far side and move all their main base activities to the base in the counter Earth position.
Earth has an average radius of 6,371.0 kilometers. If the alien base was spherical, and had a radius of 6.371 kilometers, it would have 0.001 the diameter of Earth, 0.000001 the surface area of Earth, and 0.000000001 the volume of Earth. If the alien base had the same albedo (reflectivity) as Earth, it would appear one millionth as bright as seen from Earth as an earth like planet would at that distance.
Or the aliens could make their base a hollow cylinder that rotates to simulate gravity. It might be half a kilometer in radius and one kilometer long. When the circular end was viewed it would have a area one 162,358,564th the size of the Earth's cross section, and thus it might be 162,358,564th as bright.
The hollow inner surface of the base would have a surface area 0.7853975 square kilometers. But if it had ten decks separated by 3 meters each the upper decks would not have much lower gravity than the lower decks and the surface area would be ten times greater, 7.853975 square kilometers.
A thousand such space stations orbiting at the counter Earth point would have a combined surface area of 7,853.975 square kilometers, a million would have a combined surface area of 7,853,975 square kilometers. That would be smaller than the surface areas of Russia, Canada, China, the USA, or Brazil, but larger than the surface areas of Australia, India, Argentina, Kazakhstan, Algeria, etc. etc.
If the million such space stations were arranged in a three dimensional grid of a hundred stations by a hundred stations by a hundred stations, and each was separated by a thousand kilometers from it nearest neighbors, the grid might appear almost totally transparent like space and might seem to be no more than a small concentration of space dust at first sight and maybe later sights.
1]: https://en.wikipedia.org/wiki/Coronagraph
