  (abridged) Our project endeavors to obtain a robust view of multiplicity
among embedded Class I and Flat Spectrum protostars in a wide array of nearby
molecular clouds to disentangle ``universal'' from cloud-dependent processes.
We have used near-infrared adaptive optics observations at the VLT through the
H, Ks and L' filters to search for tight companions to 45 Class I and Flat
Spectrum protostars located in 4 different molecular clouds (Taurus-Auriga,
Ophiuchus, Serpens and L1641 in Orion). We complemented these observations with
published high-resolution surveys of 13 additional objects in Taurus and
Ophiuchus. We found multiplicity rates of 32+/-6% and 47+/-8% over the 45-1400
AU and 14-1400 AU separation ranges, respectively. These rates are in excellent
agreement with those previously found among T Tauri stars in Taurus and
Ophiuchus, and represent an excess of a factor ~1.7 over the multiplicity rate
of solar-type field stars. We found no non-hierarchical triple systems, nor any
quadruple or higher-order systems. No significant cloud-to-cloud difference has
been found, except for the fact that all companions to low-mass Orion
protostars are found within 100 AU of their primaries whereas companions found
in other clouds span the whole range probed here. Based on this survey, we
conclude that core fragmentation always yields a high initial multiplicity
rate, even in giant molecular clouds such as the Orion cloud or in clustered
stellar populations as in Serpens, in contrast with predictions of numerical
simulations. The lower multiplicity rate observed in clustered Class II and
Class III populations can be accounted for by a universal set of properties for
young systems and subsequent ejections through close encounters with unrelated
cluster members.
