  This paper describes a systematic search for high-order multiplicity among
wide visual Pre-Main Sequence (PMS) binaries. We conducted an Adaptive Optics
survey of a sample of 58 PMS wide binaries from various star-forming regions,
which include 52 T Tauri systems with mostly K- and M-type primaries, with the
NIR instrument NACO at the VLT. Of these 52 systems, 7 are found to be triple
(2 new) and 7 quadruple (1 new). The new close companions are most likely
physically bound based on their probability of chance projection and, for some
of them, on their position on a color-color diagram. The corresponding degree
of multiplicity among wide binaries (number of triples and quadruples divided
by the number of systems) is 26.9 +/- 7.2% in the projected separation range
0.07-12 arcsec, with the largest contribution from the Taurus-Auriga cloud. We
also found that this degree of multiplicity is twice in Taurus compared to
Ophiuchus and Chamaeleon for which the same number of sources are present in
our sample. Considering a restricted sample composed of systems at distance
140-190pc, the degree of multiplicity is 26.8 +/- 8.1%, in the separation range
10/14 AU - 1700/2300 AU (30 binaries, 5 triples, 6 quadruples). The observed
frequency agrees with results from previous multiplicity surveys within the
uncertainties, although a significant overabundance of quadruple systems
compared to triple systems is apparent. Tentatively including the spectroscopic
pairs in our restricted sample and comparing the multiplicity fractions to
those measured for solar-type main-sequence stars in the solar neighborhood
leads to the conclusion that both the ratio of triples to binaries and the
ratio of quadruples to triples seems to be in excess among young stars. [...]
