なしに(は)/なしで(は)

1. Doing the action afterward without doing the action N that was supposed to be done first. It can be understood as "doing... without doing...".
2. Doing an action but lacking N. It can be understood as "doing... but without...".
「なしで」is a more casual form compared to「なしに」. You can replace「なしで」with「なしに」but not the other way around.
When the second clause is in negative form, like「なしに(は)/なしで(は)～ない」, these structures are similar in meaning to「なくて(は)～ない」, meaning "if you don't have the first clause, then the second clause cannot be done", but not as formal.