Splitting a string but keeping the delimiters?

So, I’m splitting a string using

string.split(/[><]/);

So this turns something like this:

This is some text with <b>formatting</b> in it.

into this:

This is some text with 
b
formatting
/b
in it.

I would prefer if I to get something like this:

This is some text with 
<b>
formatting
</b>
in it.

Because then when looping through the array, it’s less ambiguous which are tags and which aren’t.

I’ve got a few ideas I could try to do this without using split, but they are all kind of awkward, and I’m hoping there’s a simple way.