As @sebastian-simon mentioned, "split" HTML with only regular expression is impossible. The best solution is use a real HTML parser (already shipped with your browser, if you are using Node.js, you can use JSDOM).

var str = 'test <br><span>test</span> <fake></fake> <div><p>aa</p></div>8';
var container = document.createElement("div");
container.innerHTML = str; // use a HTML element to parse HTML

// If you need to work with nested tag, you should traverse childNodes and their childNodes by yourself

// childNodes included TextNode, children not.
// [...container.childNodes] convert container.childNodes to a normal array
// so we can .map over it
var elmList = [...container.childNodes];
var tags = elmList
  // if elm is a TextNode, elm.outerHTML is undefined
  // then we use elm.textContent instead
  .map(elm => elm.outerHTML ?? elm.textContent)
  .map(elm => elm.trim()) // remove whitespaces
  .filter(elm => elm); // remove empty items
console.log(tags)