Your husband will receive nothing. You, as a service member, will receive BAH with dependent based on the zip code of where your husband is residing. You don't need a lease agreement.
None of this relates in any way to filing taxes. You decide how to file your taxes and who claims whom. Are you a Reservist or National Guardsman?
If you are then when you return from your Initial Active Duty for Training, then you no longer receive BAH or any other pays except when you drill. If you are an Active Duty Soldier then after AIT you will pick up your husband and move together (or at some point and time) to your permanent duty station and then BAH will be based on that location. Don't forget, you will also receive Family Separation Pay while at BCT/AIT.
Edit: just as your friend would receive two weeks of BAH while on a regular two week annual training, while she is working part time for the NG, I believe she will receive all her benefits including BAH based on the percentage of time she works. I've never dealt with a part timer, but it makes sense.
You will receive BAH whilke in basic and AIT, if you are married. After AIT, you will only receive BAH, if you are called up for more than 30 days. You do not receive BAH for drilling or your two weeks of annual training.
If your friend is still being paid BAH, then they will catch up to her and demand the money back. They can also be criminally charged for Fraud, for not telling them that they are still drawing BAH.
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