Q:

What version of Objectivism makes this claim?

I read something many years ago, most likely in print, that I've occasionally tried to find again. It purported to be the cosmology of Objectivism or some version of it. I believe the exact wording of its first principle was:

There is a single objective reality, and we perceive it directly.

Google returns no results for this phrase. Is this statement, particularly the part about direct perception, a tenet of Objectivism or some other recognized philosophy?

A:

Direct realism
Direct or naive realism certainly assumes in its principal versions that :

Perceptual states are belief states: To perceive an object is to acquire
   a certain number of true beliefs about it that are causally received from the object
   by using one's sense organs in a standard way. (M. S. Gram, 'Causation and Direct Realism', Philosophy of Science, Vol. 39, No. 3 (Sep., 1972), pp. 388-396 : 388.)

A more nuanced view can also be formulated :

(a) in normal cases of perception, physical or
   material objects are, in a sense admittedly still in need of clarification,
   directly or immediately perceived; and (b) that the justification for beliefs
   about such objects that results from perception does not depend on ...
   inference from the subjective character of perceptual experience ... but can be accounted for in some other way. (Laurence Bonjour, 'In Search of Direct Realism',  Philosophy and Phenomenological Research, Vol. 69, No. 2 (Sep., 2004), pp. 349-367 : 351.)

By 'inference from the subjective character of perceptual experience' is meant inference from subjective sensory experience such as (if they exist) sense data or sensibilia. 
A single objective reality
There is no valid inference from either characterisation of direct realism to a single objective reality of physical or material objects - or whatever else we might directly perceive. All that direct realism implies or presupposes is that we have some direct or immediate perception of the real, whatever the real might be. It is logically perfectly possible that the objects of our immediate or direct perception do not compose a single objective reality. It may be widely assumed by direct realists that they do and if direct realism is a correct theory of perception, they might be right. But this is an independent claim, not inherent in direct realism and not even guaranteed by its truth if it is true.

A:

Since you're capitalizing "Objectivism" and asking about other "recognized philosophies" this response is specifically about how the philosophy of Ayn Rand would actually handle such a claim.  
The piece about "there is a single objective reality" wouldn't be problematic at all because this is literally the basis of Objectivist Metaphysics. The law of identity is immutable and absolute, and Objectivists would only affirm this as Axiomatic.
The second bit about "and we perceive it directly" would raise an eyebrow of any long time Objectivist because this appears at first to be a possible trap. Cautiously in the end Objectivists affirm the validity of sense perception, but this simplistic kind of statement has long been an attack vector used against pure empiricists. It is also only one piece of concept formation and though you might perceive a mirage accurately, it doesn't make the illusion of what the mirage is presenting a physical reality off in the distance. Logic and Reason play a major role in Objectivist Epistemology so while perception is "valid" it is not the end of obtaining valid knowledge about reality.

Ayn Rand called her philosophy “Objectivism” because central to it is
  a new conception of objectivity. Traditionally, objectivity has meant
  the attempt to efface the knower out of existence, so that
  consciousness can “mirror” or “copy” reality, “untainted” by any
  processing. Skeptics then bewail the possibility of man knowing
  reality, since any attempt to do so must make use of his senses and/or
  rational faculty, both of which engage in processing.
Rand challenges this whole approach. The “satisfaction of every need
  of a living organism,” she writes, “requires an act of processing by
  that organism, be it the need of air, of food or of knowledge.”
  Objectivity consists in a mind grasping the facts by the correct
  mental processes. As Peikoff formulates her view: “To be objective in
  one’s conceptual activities is volitionally to adhere to reality by
  following certain rules of method, a method based on facts and
  appropriate to man’s form of cognition.” Source

