How can an arahant still perceive without ignorance? According to the 12 nidanas, perception ultimately arises out of ignorance. How can an arahant or the Buddha still experience sense contact when all ignorance is abandoned?
The 12 nidanas are not necessarily about things 'arising' or 'created' out of ignorance. 

Unskilful things are 'created' by ignorance, such as craving, attachment, becoming, (self-identity) birth &amp; (self-identity) aging-&amp;-death. 

Neutral things are 'tainted' or 'polluted' by ignorance, such as fabricators, consciousness, mind-body, sense organs, contact &amp; feeling. 

Perception is included in the fabricators, mind-body &amp; feeling nidanas.

It follows there are two kinds of perception: (i) ignorant perception; &amp; (ii) wise perception (often called 'full comprehension').   

The quotes below may help: 


  Monks, ignorance is the leader in the attainment of unskillful qualities..SN 45.1


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  Imagine, Brahman, a bowl of water mixed with lac, turmeric, dark green or crimson dye. If a man with good eyesight were to look at the
  reflection of his own face in it, he would not know or see it as it
  really was. In the same way, Brahman, when a man dwells with his heart
  possessed and overwhelmed by sensual desire... then he cannot know or
  see, as it really is, what is to his own profit, to the profit of
  others, to the profit of both. Then even sacred words he has long
  studied are not clear to him, not to mention those he has not studied.
  SN 46.55


~


  If a monk abandons passion for the property of consciousness, then owing to the abandonment of passion, the support is cut off, and there
  is no landing of consciousness. Consciousness, thus not having landed,
  not increasing, not concocting, is released. Owing to its release, it
  is steady. Owing to its steadiness, it is contented. Owing to its
  contentment, it is not agitated. Not agitated, he (the monk) is
  totally unbound right within. SN 22.53


~


  A monk who is a Worthy One, devoid of mental fermentations — who has attained completion, finished the task, laid down the burden, attained
  the true goal, destroyed the fetters of becoming, and is released
  through right knowledge — directly knows earth as earth. Directly
  knowing earth as earth, he does not conceive things about earth, does
  not conceive things in earth, does not conceive things coming out of
  earth, does not conceive earth as 'mine,' does not delight in earth.
  Why is that? Because he has comprehended it, I tell you. MN 1


~


  On seeing a form with the eye, he isn't infatuated with pleasing forms, and doesn't get upset over unpleasing forms. He dwells with
  body-mindfulness established, with unlimited awareness. He
  discerns, as it has come to be, the awareness-release &amp;
  discernment-release where those evil, unskillful qualities cease
  without remainder. Having thus abandoned compliance &amp; opposition, he
  doesn't relish any feeling he feels — pleasure, pain,
  neither-pleasure-nor-pain — doesn't welcome it, doesn't remain
  fastened to it. As he doesn't relish that feeling, doesn't welcome it,
  &amp; doesn't remain fastened to it, delight doesn't arise. From the
  cessation of his delight comes the cessation of clinging. From the
  cessation of clinging comes the cessation of becoming. From the
  cessation of becoming comes the cessation of birth. From the cessation
  of birth, then aging-&amp;-death, sorrow, lamentation, pain, distress &amp;
  despair all cease. Such is the cessation of this entire mass of stress
  &amp; suffering. MN 38