ED has some unique aspects of it that I think complicate things more than other specialties. That said, we've been at the game a while and made most of the mistakes, so I will let you know what has worked for us personally (I WFH full time, we have two kids that are 4 and 1).
From a home perspective:
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Outsource what you can. We eventually settled on a full time nanny for childcare. She does an amazing job with childcare, but she also does a tremendous job with cleanup around the house, running errands (picking up groceries, etc.), and other things that do a ton to reduce our decision fatigue.
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Things will probably never be 50/50, but identify the sticking points for yourself and see if your spouse can help there. As the person that works from home, the mental load will probably always be disproportionately on you. For example, the nanny search and interview process fell very much moreso on me, because I'm the one that is home all the time and is the one around to effectively be the manager when things do come up. It just is what it is when one of you works from home and the other physically does not.
However, it is up to your spouse to follow through on the things they agree to cover around the house. So for me personally, I don't need my spouse to actually do 50% around the house. But if she does the ~30% I despise, I'm honestly ok with the other 70%.
3. Pay lots of attention to sleep hygiene. Given the constantly shifting schedule for EM, it can make the sleep situation hard on not only docs, but on spouses as well. Take the time to figure out what you both need to do to get adequate, restful sleep.
From an ER perspective
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Less shifts, if possible. My spouse previously worked ~140 clinical hours a month and we've managed to find a good job where full time is ~120 clinical hours a month. That has made a noticeable difference for us.
Yes, there is a slight loss of income, but we paid damn near 50% in taxes on those last two shifts anyway. So well worth the quality of life difference.
2. Finding a job where the shifts are at sensible hours.
At our old hospital, morning shift was at 6am with a 35 minute commute, and the evening shift was 3pm-1am. Both of those shifts severely messed with both of our sleep schedules.
At our current hospital, the morning shift is at 7am with a 15 minute commute and the evening shift is 12-10pm. They also have full time nocturnists. So we can keep a pretty regular sleep schedule of ~1030-630 almost regardless of my spouse's shift. I cannot tell you how big of a difference this has made for both of us (and probably more so me).
3. Make friends with the scheduler-- our scheduler knows that if they ever need someone to cover the 7am shift during the week, my wife will do almost anything to get that shift. Yes, the last minute changes are a little annoying sometimes if they need coverage last minute. But it also means she gets assigned a disproportionate number of 7a-5pm weekday ED shifts which is the fucking dream.
She still has to work weekends like everyone else, but it drastically reduces the number of evenings I work all day and then solo parent two toddlers, and number of weekend days I spend solo parenting. Our nanny being willing to work slightly later on days my spouse works evening shift, and pick up some time on the weekends when my spouse works also was a major factor in deciding on nanny vs other options (e.g. daycare).