I'm guessing that by complexity, you mean an estimate for the number of legal positions that aren't 'clearly' lost by some measure (otherwise the answer is exponential, despite the addition of your constraint), as if to build a tablebase that can play through non-trivially lost/won positions (ignoring the technicalities of the missing tablebase problem that would arise...).
One estimate of the number of 'sensible' chess games; 3^(40*2) ~ 10^40 seems reasonable; however, this answer doesn't work in this setting, because of the way your question is phrased.
I claim that our hypothetical 32-man, 'non-trivial' tablebase is composed mostly of positions arising from games that would have otherwise been drawn. I argue as follows: 60% of grandmaster games are draws; let's say (an arbitrary guess) that a third of this are agreed draws i.e. 20% of games at move 30. Another arbitrary assumption: let's say material is equal at these positions at move 30, 3 pawns and 3 pieces having been exchanged. Each side has to trade off 9 more pawns/pieces to get to bare kings. Assuming an exchange every 4 or 5 moves, again with 3 'sensible' moves in each position, the number of games concluding in this manner would be approximately 3^(30*2) * 0.2 * 3(40*2) = 0.2 * 3^160 ~ 10^80, which dwarfs the estimated number of 'sensible' chess games.
This highlights the significance of exponential growth: the percentage of
agreed draws doesn't really matter (as long as it's not zero), and neither does 'clearly' won/lost positions in the setting of a '32-man' tablebase.
My main point is, a fairly reasonable estimate to your required answer would still be of the form n^m where n is the average number of 'sensible' moves in a position, and m is the expected number of half-moves to get down to bare kings.
Also, to be precise, the number of positions (double/triple/multiple-counting transpositions) is closer to m*(n^m).
Now of course, if you include trivially drawn positions, things become really tricky, but we have to draw the line somewhere...