How does the TBS-RTS duality work? Can you make multiple moves on the strategic map, and then play them out in real-time one after the other? How much tactical control do you have over your forces in the RT battles?
TBS-Strategic Map
There's a lot to the strategic map, but it's all really easy to learn and do. If you forget to do anything, the game reminds you before you can end your turn, kind of like Civ 5 does.
So on the strategic map, you have your general(s) and your army. You can have as many generals and armies as you can afford. You decide where they go, if you want them to go anywhere at all. You can move every army every turn if you want to. Armies can be in several different stances, and each one impacts how far it can move. Stances can also impact how tired the troops are if you attack that round or get attacked during the AI's turn (all this is explained in a pop-up box whenever you are changing stances). If your army is in a region you control, you can recruit new troops based on how advanced that region is and what types of buildings you have built there. You can also combine units that aren't at full strength and dismiss units you no longer want.
Also, the strategic map is where you level your generals and your special characters. It's an easy leveling system, but pretty in-depth.
As far as your cities and provinces go, you can issue decrees and build new buildings as the populations grow. There's a chart that shows each building and the buildings that come after and tells you what each building does. Some buildings unluck particular troops, spies or envoys. Some buildings give buffs. Some give you resources each turn. Some buildings make the populace happy or unhappy. If they get too unhappy, a rebel army will appear outside the city. You've got lots of warning, though, so this almost never happens.
To improve your cities and build new and better armies, you have to manage several resources as mentioned in my earlier post.
Also, you can enter into negotiations with factions you are in contact with. You can sign non-aggression pacts, barter for supplies you need, ask them to join you in war, or even try to get them to confederate with you and hand over their cities.
That's not an exhaustive list of things you can do in the strategic map, obviously, but it should give you an idea of the sorts of things you would be doing.
RTS-Battles
First of all, except on the very highest difficulty, battles can be paused as often and for as long as you want.
So I have to talk in generalities here because I haven't manually fought a battle in Troy yet. Whenever you go to enter a fight, you have the option to fight it yourself or have the AI auto-resolve it. I've been in quite a few battles so far, but I pick my fights well, and there hasn't been any need for me to manually fight. If the game says I would win decisively, I don't fight it myself. I only fight it myself if the game says I'm going to lose or if I'm going to lose too many troops. So far that hasn't come up. Some people manually fight every battle. They are pretty fun to do.
So when the battle screen opens, all your units are spread out before you. There is unlimited time at the beginning for you to move them wherever you want within your selected area. So, for instance, you can put cavalry units on the flanks and archers in the front. I'm not sure about in Troy, but in most TW games you can also set different formations for the units, like if you want them bunched together or spread out or arranged in a certain way. Once you have everyone where you want them, taking into account things like elevation and cover (if the terrain allows it, you can hide units from the enemy then surprise them with a charge against their flank or completely encircle them), you start the battle.
Your units will do diddly squat unless you tell them to, but you can have various units set to do certain things, like having your archers fire as soon as the enemy is in range, or putting a ranged unit in skirmish mode (they'll try to avoid melee). Basically you tell each unit where to go and how fast. You tell them when to rest, when to charge, when to use any special abilities (Your hero or special character may have them), when to stop shooting, when to retreat, etc. You basically manage the whole thing.
During the battle, units take attrition and their morale changes. Maybe they are doing well and are uplifted. Maybe things are going poorly and they are thinking about turning and running. Same thing for the enemy. Their general is a mythical hero like yours, but if you can take him out, the morale of the remaining forces will be terrible, same thing if you can do simple things like flank and attack them from behind.
All the different types of units are better or worse at different things. To the best that you can, you want to put them in good situations. If they don't have armor piercing damage, don't throw them up against heavily armored enemies, etc.
And if you've never seen a Total War battle, they are amazing. Every single soldier is there doing his/her thing: fighting, running, shooting, marching, whatever. You can have thousands of soldiers on the battlefield at a time, and you can zoom in and watch what each one is doing if you want to. I love it.
There's a lot to battles, and it can be overwhelming at first, but pausing solves all problems. They are a lot of fun once you get the hang of them, and there's nothing like being far outnumbered and pulling out a victory. Or you could do what some people do and auto-resolve every single battle.