The Eidolon is a manned aircraft that functions similarly to the Phantom, albeit with many distinct features. The Eidolon is capable of firing at both ground and air units, fires in five shot bursts, and is equipped with Tachyon weaponry, allowing it to tear apart Special armor units with ease.
Eidolons are equipped with burst-firing Tachyon lasers, which penetrate deeply into Special armor, though do not inflict as much damage against Infantry targets or the Base. Their burst-fire allows them to suddenly unleash a wave of shots, allowing them to retreat or continue attacking if needed. Their weaponry can reach both air and ground units.
Eidolons have quite a bit of reach with their weaponry, the same distance as Shrikes. This allows them to outrange units like Marines, Scouts, Saints, Mirages, and Black Queens, engaging first and doing so before they enter retaliation range. Use this especially against Saints, Mirages, and Black Queens to rip them apart before they can do significant damage.
Eidolons are highly effective against Special armor, destroying any non-Roman android unit, Hover Tanks, Saints, and Mirages rather fast from the high damage output. This is especially effective against Hover Tank stacks, since they cannot retaliate back and rapidly fall apart under the consistent Tachyon rain.
Eidolons are highly vulnerable to Sakata Mk-II's, especially since they outrange the Eidolon and prevent penetrating their Special armor easily. The Mirage is also a potentially dangerous unit due to its high speed and high burst damage, making engagements with it close or losses once it reaches engagement range.
Being a mid to late-game Titanium armor air unit, Eidolons are at risk of suffering heavy damage from Special Operations and Black Queen missiles. For Black Queens, keep away from their attack range to avoid being blasted with several anti-air missiles, instead poking from a distance to break down the Special armor.
Marine and especially S-Marine groups can be troublesome for the Eidolon, since it only barely outranges them and doesn't even do significant damage to them. It also is not that effective against the Base either, making it hovering close to it risky in case of sudden counteroffensives.
With enough Influence and resources in hand, you can make a few Eidolons, stack them, and have them ready for use against Hover Tank and late-game air units. They'll be more effective than just one at breaking down Black Queens and Saints before they become an issue, and against single Mirages, they'll destroy them before too much damage is received. You can use this to bolster a pin as well.