Ok, after playing a bit more with battle timer off, I think I realize now how the reinforcements work. It's not based on time (i.e. a random amount of time is chosen for your reinforcements to show up after you drop below 15 units), instead there is a chance to trigger reinforcements every time one of your units leaves the field. It might be predetermined (i.e. the random choice is "this game, after your first 5 units leave, reinforcements will show up"). Or it might be a fixed chance per unit (i.e. every unit leaving the field will have 10% chance to trigger reinforcements). I wouldn't know.
So, rather than simply withdrawing units (empty archers mostly) and waiting for reinforcements, I've learned to withdraw more units to trigger the reinforcements, if nothing shows up right away. Choose whichever are heavily damaged or very tired, or otherwise least useful in your situation. If any unit you withdraw is a big loss, withdraw one at a time, so you don't end up withdrawing more than necessary.
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