Ptolemy's second book divides the inhabited world into four quarters and puts each in familiarity with one of the four triplicities of signs. Asia is the only landmass in the scheme that falls across two of them.
The southeast quarter, which Ptolemy fills with India, Persia, Babylonia, Mesopotamia and Assyria, is in familiarity with the earth triplicity, Taurus, Virgo and Capricorn. The northeast quarter, which he calls Scythia and populates with the lands north and east of the Caspian, is in familiarity with the air triplicity, Gemini, Libra and Aquarius. Ptolemy assigns planetary co-rulers to each quarter as well, and those assignments are not identical to the ordinary triplicity lords, so this entry does not name them rather than report one as the other.
Splitting Asia is not a failure of the scheme. The scheme is not a map of continents. It is a division of the oikoumene, the inhabited world as a second century Alexandrian understood it, by compass quarter from a centre near the Mediterranean, and continents are simply not what it divides. Reading it as geography produces the impression of an error where there is none.
What the quarter assignment is used for is mundane astrology, the reading of eclipses, comets and great conjunctions for their effect on regions. Ptolemy's chorography is the lookup table that tells such a reading which part of the world an event in a given sign concerns.