Thanks for contacting Just Answer - Let's see if I can help here...
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OK..... I am really glad you contacted me with this specific question because I have dealt with this issue many times as you know about 50% of the families in the U.S. are "blended" for one reason or another just like yours.
Having said that, I can tell by what you have posted that you know how to be a good parent!! :) I must tell you that many times I have to start from square one in terms of building relationships, setting up boundaries and etc.
Now here is the core response that I believe will help - it may not be what you want to hear right away, but I can tell you that this is what has helped so many other step families and I will further assert that almost all experts agree with this approach......
I'm not suggesting that your husband is right - HOWEVER....... in some regards XXXXX XXXXX a point (please stay with me - I really want to help).
Anytime a step parent comes in and tries to assume the disciplinary responsibilities, it can turn into a no win situation for all involved. This happens for a number of reasons and you are a first hand witness to some of that now (based on your post of the kids telling you that their "real" mom would never make them do that). While this may or may not be true, there is no response that you can give that will be adequate, and in fact if you respond at all you are likely to let them take you "down a rabbit hole". And as I said, nobody wins here.... you don't get the respect you (richly) deserve, the kids don't get the discipline they need and if fact may end up resenting both you and there mother (remember these are kids and they don't always think rationally).
So, what to do?
Well, here is the only way to make it work - no, it won't be easy, but it will work and what you are doing now is not working and could potentially get worse :(
Here is what you need to do... you need to support your husband in HIS efforts to discipline the his kids in the same way he need to support you in the disciplining of your kids. Yes, I heard you say that he is not much of a disciplinarian - I definitely heard that - however...... That is not an option any more!
Now, here is how it really works. You two must agree to be "allies" for the love not only of each other, but for the love of all the kids. Then you two develop a plan of action as to what the expectations are.
Once you agree on this plan, he explains it to his kids and you explain it to yours. You let them know the expectations and the consequences for completing their tasks (positive reinforcement such as going to the movies, etc) as well as the consequences for not completing their tasks (losing privileges, etc).
Since you seem to be the one to take charge of the home for matters such as this, there is good news :)
When you and your husband create you plan behind closed doors, he can let you take the lead - the kids don't have to know this - it will be between you and him. Just present it as a plan you both created and present it with a unified front.
Also, part of the idea is that he says to his kids, I am in charge of raising you. If you have problems you can come to me, but understand that I will consult with my wife as she is my support. Also, even though I am in charge, I expect you to show her total respect as she deserves it.