Had to take a break to do something with my wife, but that still took longer than I thought...
For the battle I represented Andronikos' two depleted units with a single Byz cavalry unit (it was a hard choice. The Byz cav had slightly higher def but a lower missile attack, as opposed to the Turkomans. I saw them as basically equal, though). I lowered the experience from 3 chevrons (which both depleted units had, amusingly) to 1 and did not include the +1 armor the depleted Byz cav had.
Phase One was the missile duel. I played it twice from Andronikos' side and twice from Theo's. There was an anomaly in that I could not go into a cantabrian circle (the button would light up then turn off...) which may have affected Andronikos' losses. I did not use loose formations for any units as none were specified, although I did take the foot archers off of skirmish while playing Andronikos' side, as Theo had given no orders to try to distract them by moving his HAs close enough to make the skirmish (the AI on the other hand would have for a least one battle). Also, I wanted to use 5 volleys exactly as the length of time for the missile duel before attempted melee. In both tests I ended the battle after what I thought to be five volleys by the footarchers, although I am less certain about the second one (which might explain losses of about 6-8 more units on each side).
In both tests Andronikos' missiles slaughtered a Skythikon unit, destroying or nearly destroying it, plus averaging about 12-20 more kills from other "Fatimid" units. Both tests from Andronikos' side showed fairly minimal losses, although casualties on both sides were higher in the second test.
I then played the missile duel twice as Theo. As suspected there were somewhat fewer losses for his side while I was playing it, although I did not do anything special that would affect the results. Surprisingly the AI footarchers still managed 5 volleys without much reforming, thankfully. While playing as Theo I was able to heavily damage one Skythikon unit, taking losses about equal to a single unit of men myself. Test two saw slightly more losses for Andronikos, about the whole unit of Skythikons.
Interpretation: I decided that Theo lost one whole unit's worth of men at least and deleted his Skythikon unit. It might have been fair to deplete one of his other units a little, if there were an easy way to do so. I also deleted one full unit of Skythikons for Andronikos. His losses really tended to be less than a full unit, but given that he'd gained twenty more horse archers (at the cost of a couple chevrons and armor, but still) it seemed fair.
I played around seeing if Andronikos' men could avoid a charge. I eventually decided no. Theo's remaining Skythikons were swift enough to catch just about any cavalry units, after a couple tests. Not being able to test while using Cantabrian circle was annoying, but since that would make it harder for Andronikos' men to flee (it takes a second to get out of) I figured it didn't matter.
For
Phase two I played both sides twice again (one of them four times, actually, but we'll get to that). Since Theo's men were doing the charges I figured they would remain within range of the footarchers, and so included the latter in the tests to take what potshots they may.
Both times as Andronikos he destroyed Theo's cavalry, but only after horrendous losses to both HA units of his, leaving him with maybe 2/3ds of a unit. I should note for this I had to charge Andronikos' men into Theo's, bringing them far away from the foot archers, which wouldn't happen with Theo doing the charging.
Playing as Theo had surprising results in that he came out worse with me playing him. On test one I had each of Theo's units charge a HA unit of Andronikos' (the AI kept them separate). Andronikos only lost 40 men while Theo lost all of his men. The footarchers continuing to fire made a big difference.
Trying again that way things went a little better for Theo, with Andronikos losing a whole unit's worth.
I then tried two tests with Theo's cav both charging the same unit of Andronikos, only going to the other after routing the first. This went better, with Andronikos ending with less than a full unit over all, and Theo's horses destroyed.
Given that while playing phase two I wasn't quite adequately representing Theo's likely losses (1 1/2 units or so) if I were asked what the final results should be I'd say all of Theo's units disbanded, with Andronikos retaining his footarchers and one unit of Byz cavalry, or better yet an almost full unit, 40-50 or so (although that would be a pain to simulate.

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I have screenies if needed.
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