How to vertically middle-align floating elements of unknown heights?

I have a (horizontally) centered outer div containing two elements of unknown width:

<div style='width:800px; margin:0 auto'>
  <div style='float:left'>...</div>
  <div style='float:right'>...</div>
</div>

Both floats are top-aligned by default, and are of varying/unknown and different heights. Is there any way to make them vertically centered?

I eventually made the outer div

display: table

and the inner divs

display: table-cell;
vertical-align: middle;
text-align: left/right;

but I'm just curious if there's a way to do this with the floats.

You can't do this directly, because floats are aligned to the top:

If there is a line box, the outer top of the floated box is aligned with the top of the current line box.

The exact rules say (emphasis mine):

4. A floating box's outer top may not be higher than the top of its containing block.
5. The outer top of a floating box may not be higher than the outer top of any block or floated box generated by an element earlier in the source document.
6. The outer top of an element's floating box may not be higher than the top of any line-box containing a box generated by an element earlier in the source document.

8. A floating box must be placed as high as possible.

That said, you can take advantage of rule #4:

• Place each float inside inline-level elements that establish a new block formatting context /BFC), e.g. display: inline-block.
• These wrappers will contain the floats because they establish a BFC, and will be one next to the other because they are inline-level.
• Use vertical-align to align these wrapper vertically.

Be aware that some space might appear between the inline-block wrappers. See How to remove the space between inline-block elements? to fix it.

.float-left {
  float: left;
}

.float-right {
  float: right;
}

#main {
  border: 1px solid blue;
  margin: 0 auto;
  width: 500px;
}

/* Float wrappers */
#main > div {
  display: inline-block;
  vertical-align: middle;
  width: 50%;
}

<div id="main">
  <div>
    <div class="float-left">
      <p>AAA</p>
    </div>
  </div>
  <div>
    <div class="float-right">
      <p>BBB</p>
      <p>BBB</p>
    </div>
  </div>
</div>

Another approach would be to use flex -- it could be a replacement for a float if you have two parts. One (floating) would be of auto size, and the second would grow to occupy entire container. On the cross axis you pick center and voila, you have the effect of the float and centered elements.

Here is beautiful cheatsheet about flex: http://jonibologna.com/flexbox-cheatsheet/

Nope this is when table cells suddenly seem like a great idea. If it is a fixed height you can use line-height.