Using the Goalkeeper to Break the Press
Football solved a version of this problem by pushing the goalkeeper into build-up play. Water polo's shot clock rules out a direct copy, but not the underlying idea.

One of the clearest tactical developments of recent years is the rise of high pressing, particularly from the Spanish national team. The aim is to stop the ball reaching dangerous areas of the pool where a pass to the pit or a shot on goal becomes possible. The problem for the team in possession usually isn't that the ball can't get there. It's that by the time it does, there's no time left to do anything useful with it.
Learning from football
The obvious question is how to play against a press like this. No press is unbeatable, but having more than one tool to break it matters. Football offers one solution I haven't seen used in water polo: goalkeepers actively involved in build-up play specifically to break a high press. Using the goalkeeper creates a numerical overload. If every presser is assigned a man, there's always one attacking player left over.
Water polo has a real constraint football doesn't: a shot clock on every possession. A tactic built purely around keeping the ball, only to lose it 28 seconds later anyway, isn't viable, and the trend of shrinking attacking clocks suggests the game's lawmakers don't want it to become one either.
What it could look like
None of that makes the press less effective in water polo, even if it shows up less often than its football equivalent. But a goalkeeper could, in principle, push up toward the halfway line in the opening moments of an attack, purely to break the press and create an extra outlet. Two problems follow immediately.
First, a team could simply leave the goalkeeper unmarked and press everyone else as normal. But that isn't free. Any player at a high level can score from two metres, so no team lets a goalkeeper advance that far unchallenged. The moment someone steps out to close the goalkeeper down, the press is already broken, because a pass into the pit can now be made from five, six, even seven metres out, without pressure on the ball. Pressure on the passer is what makes an entry pass hard in the first place; take that away and the press has done its job of forcing a decision but lost its point. The goalkeeper's role here is only ever to force that moment. Once it happens, a settled defensive shape can be set up against teams that would otherwise rather avoid these situations altogether, or delay them as late as possible.
Second: what happens to the goal if possession is lost while the goalkeeper is out of position? My answer is that the goalkeeper starts retreating the moment the press is broken; the real risk applies mainly to turnovers, not the whole possession. And a turnover here isn't as costly as it looks. If the opposition had set up to press, that same shape converts instantly into a press for the other side once possession changes hands, because the marking is already man to man. A shot from distance in that moment risks wasting a possession entirely. Most players can put the ball in an empty net from range, but doing it while also being closed down is a different proposition, and it's not obvious that ends in a goal often enough to be worth it. Especially late in a game, a team that knows it needs more possessions to change the scoreline may find it harder to justify that shot, knowing the price of missing is handing the opponent a renewed possession of their own.