Stiffen precious material, naturally preserved for Tudor court (4,7)
I believe the answer is:
star chamber
'tudor court' is the definition.
(English court of the Tudor era)
'stiffen precious material naturally preserved' is the wordplay.
'stiffen' becomes 'starch' (starching is a kind of stiffening).
'precious material naturally preserved' becomes 'amber' (amber is a natural resin).
'starch'+'amber'='STAR CHAMBER'
'for' is the link.
(Other definitions for star chamber that I've seen before include "Old-fashioned court" , "Old court" , "Court abolished in 1641" , "Tyrannical court" , "tribunal" .)