"equally-sized" (adv + adj) does not require a hyphen.

"equal size water pools" (adj + noun + noun + noun) needs a hyphen.

So that leaves us a choice between the second and fourth renderings.

I prefer the fourth one:

• . . Two [water] pools of equal size.

simply because it does not have three consecutive nouns.

(However, I put "water" in brackets, because I feel it is not needed. Unless specified, a "pool" is assumed to be a pool of water—even in a desert!) If you take out "water", #2 sounds just as good as #4.

COCA shows enough results for 'equal-sized' that serve as an adjective. So, in your case, it could be...

There is a desert with two equal-sized water pools.
We then found cutoff points within each category to form 3 nearly equal-sized groups (COCA).

or, another way to say this is the last option you quoted. Use 'of equal size' after the noun in concern.

There is a desert with two water pools of equal size.
Break the dough into 15 portions of equal size (COCA).

The words 'equal-size' is also possible but less frequently shown.

Fetch the dough from the refrigerator and cut it into eight equal-size pieces (COCA)

Finally, 'equally-sized' is an improper use of those words. Don't use it.

'Equal size' without hyphen is also found. Again, less frequently used.

By cutting between the uprights, I can get four equal size pieces of firewood (COCA)

So, to sum up:

There is a desert with two equal size water pools -less frequent
There is a desert with two equal-size water pools -less frequent
There is a desert with two equally-sized water pools -improper
There is a desert with two water pools of equal size -proper (and preferred?). And adding...
There's a desert with two equal-sized water pools -proper (and frequent?).

I would rather go with:

There is a desert with two equal-sized water pools.[In place of 1st sentence]

OR

There is a desert with two water pools (that are) equal in size. [In place of 4th sentence]

I'm not a native speaker, but I find above sentences correct. IMO, 'equally' cannot be used before size or sized.