How to define an independent interference by God? The most clear evidence when people state their belief is experience.
The reasons vary:

logical expressions of belief is somewhat inverted (read this paragraph)
experience is connected to intimate feelings and thus valued over analog reasoning
maybe others...?

During my studies of calculations in probability at the university, I had to be very careful to distinguish events and their relative dependance on each other.
If you do not rightly separate events from their dependencies -  the calculated results can be very surprising because intuition can be misleading. If you are not very familiar with this, there exists a very basic problem that illustrates a simple example: Monty Hall problem
&quot;Two Events are independent...
...when they have no connexion one with the other, and that the happening of one neither forwards nor obstructs the happening of the other.”
I have strong problems of regarding so called personal experiences of god as true evidence of his interference in life because they are not independent. They cannot be verified as being divine in any way.
Usually, the experience...

...can be explained to be a rather unlikely event, but that does not make it divine.
...is only real to the witness. (narrator; one person)

In the last case the experience is not an independent event to a third person like me. The experience is always connected to the witness.
What would true independent divine interference 'look' like?
How can you define divine interference?
Your question makes the assumption that some good things occur as the result of divine interference, and that others occur by way of chance.  This is a faulty assumption.


  Every good and perfect gift is from above, coming down from the Father of the heavenly lights, who does not change like shifting shadows. - James 1:17 NIV


So in answer to your question, what would true independent divine interference 'look' like?:

It would look like a good and perfect gift.

Now I know your question isn't asking specifically about "good things," but more generally about determining whether "personal experiences" are divine.  I think the same litmus test can be used, though.

Was the personal experience a "good and perfect gift?" If so, it came from God.

Any genuine personal experience with God also won't contradict any other part of scripture.

I believe most protestants (and possibly other groups) value personal experience, but always in light of scripture.  Personal experience is considered a very real, and valuable thing, but without an ironclad standard (God's word), it is too easy for it to contradict reality (and even for one experience to contradict a later one, experienced by the same person).