Zara Tindall is “very close” to her brother’s girlfriend, according to new claims. Peter Phillips, Princess Anne’s son, debuted his new girlfriend, Harriet Sperling in July. The NHS nurse is now said to be “close” with both Zara and her husband Mike Tindall. Mike was left on parenting duties as he watched Zara competing at the Burgley Horse Trials. The 45-year-old was joined by Zara’s brother and his girlfriend, Sperling. Onlookers claimed Mike was in “full-on dad mode” as he kept an eye on youngsters Lena and Lucas. Eldest child Mia, 10, was allowed to roam the site more freely, equipped with…