The best replacement for presence is presents.

The best alternatives to presence are rated as following:
★★★★★  presents
★★★★  absence
★★★  presence
★★  living tribunal
★★  attendance
★  toaa
★  treble
★  unholy presence
★  one above all
★  immersion
★  primal monitor
★  beyonder
★  transubstantiation
★  blood presence
★  hologram
★  superman prime
  absence of
  present
  bona fide resident
  unholy
  existence
  continuous presence
  real presence
  prejudice
  unlawful status
  frost presence
  god's glory

Your alternative for presence is not listed? Suggest!
Speaking about presence, one probably shouldn't forget about presents.

Other search terms related to presence and their ratings are:
★★★★★ presents
★★ us
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Google Trends

Scale is based on the average worldwide traffic of presence in all years. Click on graph for more.
presence
presents
absence