Those inclined to vote Tory would have soldier turning up in tatty T-shirts (9)

'those inclined to vote tory' is the definition.
(the Conservative or Tory Party is right-wing)

'soldier turning up in tatty t-shirts' is the wordplay.
'soldier' becomes 'GI' (GI can mean a US soldier).
'turning up' shows that the letters should be reversed in order.
'in' indicates putting letters inside.
'tatty' indicates an anagram (I've seen this in other clues).
'gi' written backwards gives 'ig'.
't'+'shirts'='tshirts'
'tshirts' is an anagram of 'rhtists'.
'ig' put into 'rhtists' is 'RIGHTISTS'.

'would have' is the link.