1. What does it mean?
본계 shortens 본계정, the account someone treats as their principal online identity. It contrasts with additional accounts used for other interests or purposes.
Also written or said as 본계정, bongye
2. The nuance
It does not mean a government-verified or officially certified account. A person's main account can still be private, and the meaning is unrelated to other dictionary senses of 본계.
3. Examples
“중요한 공지는 본계에 먼저 올릴게.”
I'll post important announcements on my main account first.
“부계 알림을 꺼 놔서 본계로 연락해 줘.”
I've muted notifications on my secondary account, so contact my main one.
Sources
References support the meaning and usage. Example sentences are written for NoCapWiki.