My previous employer used to fire employee, if they find they are doing second job or even 1099 with out written consent. It was baked into their employees rules of behavior.

For my current employer, we need consult with HR before taking second job. Saying that I have seem, mostly Indian taking 2 full time jobs on and off and lot of americans do 1099 or consulting through their registered company.

In summary, it's hard to justify commitment when you have two full time job, specially when you work remote. It's definitely not illegal, but might breach some company policy. If you want to do it, I would recommend to work on 1099 and timing fexiability written out in job offer, so that you can cover your base if something goes bad.