hannah-dulset:

‘다가 보니까’는 동작동사에 붙어 어떤 일을 여러 번 계속하거나 상황이 지속되어 어떤 결과가 생겼다는 것을 나타낼 때 사용한다.

‘다가 보니까’ is used
to indicate that the speaker learned something new or a certain situation has
occurred as a result of doing some action or behavior continuously/repeatedly
from a point in the past.

*Can occasionally be used with 상태동사

다가 보면     vs     다가 보니까

다가 보면- Is used when discussing a situation involving general knowledge. Because it is also stated hypothetically, the second clause can only be in future or present tense.

자주 만나다가 보면 친해질 거야.
If you meet (the person of conversation) often, you will become close.

다가 보니까- is used when discussing a situation involving personal experience. Therefore, because it is something the speaker has already experienced, the second clause has to be in past tense.

자주 만나다가 보니까 친해졌어.

Since we (the speaker and the person of conversation) meet often, we’ve become close.

고 보니까     vs     다가 보니까

고 보니까-

• Occurs after an action is complete
• Refers to a time after performing an action once.
• The following clause contains new/unexpected information learned by the speaker.

다가 보니까-

• Occurs during the process of an action
• Refers to a time during the process of performing an action multiple times.
• The following clause contains newly learned information/state of affairs resulting from the preceding action.

Korean explanation from Kyunghee University textbook, English/translation my own. Any questions or correction please let me know~