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The Tagma doesn't actually have combat stats that are any better than the Knight it replace; all it does is boost adjacent units. The threat comes from other factors.
- Byzantium's trait grants bonus combat strength to everything; usually +3 to start with for their founded religion, but it can snowball if they go on a conquest and conversion spree.
- The Hippodrome and its Arena building grant free heavy cavalry units, so there will be a lot of Tagmas. The reduced resource cost also makes it easier to amass them.
- If a Tagma does decide to attack your walled city directly, it'll do full damage thanks to Basil's trait (if that city has been converted), rather than the greatly reduced damage everyone else's cavalry does.
That said ... the AI lacks that killer instinct, and can often be discouraged from actually attacking even when you couldn't hold against it. Back in the Classical era of my current game (playing as Basil, as it happens), after Kongo declared war against me while most of my forces were on another front to attack Portugal, they sent a large force to a border city. No walls, just an archer, a hill, river crossings in every direction, and Victor (first promotion, which boosts garrison strength). But against three of their Ngao Mbeba sword-replacement UUs backed by my Crusade belief (Mvemba's terrible trait actually mattered for once), that wouldn't have been enough to hold. They had a strength advantage on the attack; all they had to do was attack with all three, two turns in a row.
But they didn't attack. They just walked around the city for several turns while I chipped away at their health and built the wall. In the end, they tried to attack my vassal city-state, but lost the entire force as my ranged attacks whittled them down and their target put up walls as well.
The lesson? The AI doesn't like to attack hard targets. Harden your cities with walls and ranged units, and they'll mill about trying to set up an attack or pick off something vulnerable farther away rather than actually committing, letting you chip away at their health until they die or are forced to retreat.
The first unique that makes Basil powerful in combat is that he gets the latest heavy cav version for free with every entertainment center and its buildings he produces. He therefore gets a free quantitative advantage over you, also qualitative in the sense that he is incentivized to beeline for the next heavy cav version. This is the factor in Basil's strength over which you have no control, well, aside from this, that you can prioritize taking his cities as early as you can. Taking away his cities is the only way you have to keep him from making entertainment centers and their buildings.
The second set of Basil uniques motivates you, at least when you are on defense, to take the passive step of avoiding open field combat as much as you can, and relying on your walls. The tagma does have as its only advantage over the knight that it confers a combat bonus on adjacent units, so there's that advantage to open field combat that Basil gets from having tagmata around. And, given the quantitative edge he's likely to have because he gets to print units, there's going to be quite a bit of this adjacency available to him.
Thirdly, the most powerful unique Basil possesses for conquest is that his heavy and light cav get to ignore walls of a city that follows his religion. He gets to print units, fast powerful units for open-field combat, and, unlike any other civ, his cav that he gets to print have an ability to just ignore walls, the hugely important aspect of static defense. Basil gets a freebie equivalent of the amazing Akkad suzerainty bonus, for cav instead of melee and anti-cav, so he is free from the need for either of those unit lines. The Achilles heel here is that your cities have to follow his religion for him to be able to get this advantage. This means that the most important active measure you can take to defang Basil is to hit his religious expansion, rather than purely combat measures.
One way you keep Basil from converting your cities is to keep him from killing your units, because yet another of his uniques is that he gets religious conversion pressure from nearby kills. This reinforces the second point, the need to mostly avoid open-field combat with Basil. At least on defense, when he is attacking you anywhere near your cities, come out of your walls and fight him in the open only when you can kill his units with a high degree of certainty, without losing many or any of your own.
Beyond that, you may need, or at least find very useful, religious combat. Condemn any religious units of his he sends to your territory. If you are already at war, doing this can be tricky because his military will then get a chance to kill the unit you used to do the condemning. Having inquisitors handy to deal with any conversion he does manage to accomplish is useful.
Best of all is prevention. If you start anywhere near Basil, taking all his cites is the best preventive, because it takes away his ability to get both entertainment districts and holy sites. If conquest is not practical, taking away his religion with your religious combat units is second best. Converting his holy city takes away his +3 combat strength. Converting all his cities takes away his ability to build religious combat units, plus passive conversion pressure on other cities.
The other dimension of the religion/combat synergy that Basil uses, are the particular beliefs he chooses.
Crusade is a natural fit for him, because he gets +10 combat strength to add to your woes, in cities of yours that follow his religion. This synergizes enormously with his ability to ignore walls in your cities that follow his religion, so if he does have this belief, it becomes paradoxically advantageous for you to do as much fighting as possible in his lands rather than yours. This is tricky because that involves sending your units out in the open where their loss gives him conversion pressure, but that will only come back to bite you if you can't keep the upper hand in open-field combat, and Basil is able to turn the tables and invade you.
Basil might instead choose Defender of the Faith, which gives him +5 in the territory of his own cities that follow his religion. That's nice for you because this belief is a lot less of a threat than Crusade, but it does invalidate the incentive to fight on his territory. You still do that if you can, because taking his cities is the very best way to undercut his amazing offensive capabilities.
To summarize: the best defense against Basil conquest is almost always an offense to fight him in his own lands and take his cities. This preventive war should be undertaken as early as you can manage -- ideally before he gets all the elements of his war machine built and in place. If forced to defend because you don't have open-field supremacy, avoid letting him kill your units, hide behind your walls more than you would against any other opponent. Fight his religious conversion at higher priority even than directly fighting his military conquest, because his religion in your cities is the source of his unique military strength.
All of this is to say that Tagma are incredibly overpowered. Pike and Shot are the best counters but don't arrive until the Renaissance Era. You either need to beat them to a religion and deny them Crusade or run them over in their territory before they get going.
Edit: if playing against AI, they will never fully utilize their bonuses. Run them over with better units and tactics.
Oh and upgrading experienced units ASAP is always a good idea
If I were to have to face the Tagma there is a very simple device for defence that negates almost all the advantage of the Tagma or virtually any of the 'Cavalry' cultures. The Caltrop. It can be made by any blacksmith up to a 100 a day. (don't try it on the Mongols) One of the issues with the game is its strongest point, the time scale.
for every weapon there is a counter, a defence, the game has however only one defensive device to offence until late in the game. The WALL. With a ranged unit sequestered behind it, very hard to crack. There are no defensive preparations to use against a larger or more powerful enemy force because we play at a "Macro level". Still in real life the Turkoman did crush the Byzantine by being more innovative, more determined and more dedicated.