Wife and mother issues

Good day mamas, please advice is needed as I am having wife and mother issues. I have been married for 10 years but we have always lived with with my parents in their house. We currently cant afford to buy a place or even rent due to paying debts and earning little money so have no choice but to stay at my parents. The problem is my wife and mother don't get along well. There's no swearing or direct insults involved but only just little irritations that gets on my wife's nerves that causes arguments between us. When i say little i mean little. For example, if my wife puts on the washing machine and goes shopping, my mother may take the clothes out when it has finished and hangs them out to dry. Rather than seeing that as a helping her, my wife sees it as a threat and kinda undermining her. We have such a big argument over these kind of small issues that sometimes we feel like splitting up. I tried talking to my mother but all she says is that she cant always know what will cause irritation although my wife says it is deliberate.
Although my mother do things that can irritate the wife, I don't think they are big enough to make a big deal. I just keep telling my wife to ignore but she does not get it. I know some women take these little irritations from other women seriously but if u look at what the chores my mother does, any human being with any sense will ignore her irritations: For example, my mother does-
- Most of the cooking for the family
- Goes out on her own to buy food shopping using her OWN pension money
- Cleans the house and clothes
- Looks after our young daughter while we go out to work
Although we contribute by paying electricity bills, the rest is covered by my parents. We don't even pay rent. So am i right in telling my wife to ignore my mother's irritations considering the amount of work she does? I am so frustrated coming home everyday and having my wife arguing with me about what my mother did that day. Hopefully in the next 2 years we would have saved enough money to move out but until then my head with explode. I'm like a teacher between 2 arguing kids. Please provide an advice for a solution?