The two countries (or one country, if you prefer) broke apart almost 60 years ago, and until this week it was not even possible to travel directly between them: Taiwan-China flights had to go through Hong Kong, and ships had to stop off en route at the Japanese island of Okinawa.
The 180km-wide Taiwan Strait remains one of the most heavily militarised regions in the world.
Even under the new government of President Ma Ying-jeou, which is committed to improving relations with the mainland, Taiwan keeps its defences up.
Taipei recently signed its largest arms deal with the United States, agreeing on a $US6.5 billion ($NZ11.5 billion) package of guided missiles, attack helicopters and other advanced weaponry.
Beijing retaliated by cancelling a series of scheduled meetings between Chinese and US generals - but it did not cancel the visit of Chen Yunlin, the most senior communist official ever to set foot in Taiwan.
Mr Chen is not formally a member of the Chinese Government, because Beijing sees Taiwan as a renegade province, not a legitimate state.
He is officially the head of a non-governmental organisation called the Association for Relations across the Taiwan Strait, and his host, Chiang Pin-kung, is the chairman of a similar Taiwanese NGO, the Straits Exchange Foundation.
But that is just a charade to save everybody's face: this is really a serious encounter between two governments.
The first results of the encounter are already known: in future, cargo ships will be allowed to sail directly between Taiwanese and Chinese ports, and there will be more than a hundred direct flights a week between cities in Taiwan and China.
There are hopes, especially in Taiwan, that this will lead to greatly increased trade between the two sides, and the next round of talks will focus on closer financial ties as well.
But where is all this leading? Reunification? The opposition Democratic Progressive Party (DPP) in Taiwan fears so, and a million of its supporters demonstrated against the meeting across Taiwan last week, but President Ma swears that he will make no moves that compromise Taiwan's sovereignty.
Well, then, could there be a permanent two-state solution in which Beijing and Taipei recognise each other as legitimate governments of independent countries? Beijing's leaders would rather die in a ditch, and so would many ordinary Chinese for whom the unity of the motherland is sacred.
The truth is that neither side really knows the destination of this voyage, but they are nevertheless setting out together.
There have been great changes in China, where prosperity has soared and the ruling Communist Party has scrapped most of its ideology over the past quarter-century, but Taiwan has changed even more.
Sixty years ago, after all, the Nationalist Party that ruled the island for so long was almost identical to the Communist Party in its structure, its nationalism, and its authoritarian style.
Both parties were formed in the wave of nationalist fervour that swept China after the 1911 revolution overthrew the monarchy, and Chiang Kai-shek, who led the Nationalist Party for 50 years until his death in 1975, was just as autocratic as his great rival Mao Zedong, the leader of the Communist Party.
But the Nationalists lost the civil war in 1949 and withdrew to Taiwan, where American sea-power prevented the communists from following, and so Taipei became the seat of the government-in-exile of the Republic of China.
That, at least, was how Mr Chiang saw it, and he harshly suppressed any expressions of Taiwan separatism. His dream was to return to Beijing in triumph as the leader of a reunited China.
But in the quarter-century after Mr Chiang's death, the Nationalist Party in Taiwan, while remaining dedicated to a united China in principle, gradually moved towards a fully democratic system - and so lost power in 2000 to a separatist party that wanted to declare an independent Taiwan.
There was genuine support for that goal in Taiwan, especially in the south, but it was never a real possibility: Beijing made it clear that a declaration of independence would trigger an invasion.
So, after eight years of economic stagnation and growing corruption, the separatist DPP lost power in last March's elections, and the Nationalists returned to power under Mr Ma.
They remain committed in principle to the reunification of China, but not under a communist dictatorship.
Improving trade with China is very important to Taiwan, which has not done well economically in recent years: the average Taiwanese still earns about five times as much as the average mainland Chinese, but the gap is narrowing fast.
However, closer political ties are more problematic, and the military still stands ready on both sides of the straits.
The two governments may be setting off on a voyage to nowhere, but at least it has started well.
Gwynne Dyer's new book, Climate Wars, has just been published in New Zealand by Scribe.
