What is the strand specificity of a reference genome?
First of all, the reference genome strand specificity is referred to as sense (positive strand) or antisense (negative strand). Now let's consider to sequencing data or FASTQ files. When we align reads, a resulting SAM or BAM file has a column specifying strand information, we usually see a + or - strand.

For more background on the strand names here's a somewhat incorrect note from the wiki article under sense (molecular biology)


  The strand names actually depend on which direction you are writing
  the sequence that contains the information for proteins (the "sense"
  information), not on which strand is on the top or bottom (that is
  arbitrary). The only real biological information that is important for
  labeling strands is the location of the 5' phosphate group and the 3'
  hydroxyl group because these ends determine the direction of
  transcription and translation. A sequence 5' CGCTAT 3' is equivalent
  to a sequence written 3' TATCGC 5' as long as the 5' and 3' ends are
  noted. If the ends are not labeled, convention is to assume that the
  sequence is written in the 5' to 3' direction. Watson strand refers to
  5' to 3' top strand (5' → 3'), whereas Crick strand refers to 5' to 3'
  bottom strand (3' ← 5').[4] Both Watson and Crick strands can be
  either sense or antisense strands depending on the gene whose
  sequences are displayed in the genome sequence database. For example,
  YEL021W, an alias of URA3 gene used in NCBI database, defines that
  this gene is located on the 21st open reading frame (ORF) from the
  centromere of the left arm (L) of Yeast (Y) chromosome number V (E),
  and that the expression coding strand is Watson strand (W). YKL074C
  defines the 74th ORF to the left of the centromere of chromosome XI
  and denotes coding strand from the Crick strand (C). Another confusing
  term referring to "Plus" and "Minus" strand is also widely used.
  Whatever the strand is a sense (positive) or antisense (negative), the
  default query sequence in NCBI BLAST alignment is "Plus" strand.


Although it is correct when it says 


Watson = Sense = Plus Strands
Crick = Antisense = Negative Strands


A satisfactory answer/convention is provided/suggested in this PMC article covering the terminology of the Watson and Crick strands 

Below are some interesting excerpts from the publication:


  The earliest reference that we could find to the "Watson strand" and
  the "Crick strand" is somewhat tongue-in-cheek and comes from a pair
  of papers in 1967 by Wacław Szybalski and colleagues. They
  bound the two DNA strands of phage λ to the synthetic polynucleotide,
  poly(IG), which has an affinity to cytosine-rich regions. They then
  separated the two strands by density, which turned out to be
  determined by the amount of bound poly(IG). In a cesium-chloride
  density gradient, the strand with more bound poly(IG) was denser and
  heavier than its complement. Because the "dense" strand was
  cytosine-rich, Szybalski and colleagues called it the "C strand."
  Logically, thus, the complementary strand, which was guanine rich,
  should have been the "G strand." Instead, it was christened the "W
  strand."


So the authors note that the Strands started on their terminology journey with their roles reversed with the Crick strand specifying the heavy strand (with IG) and the Watson strand specifying the lighter one.

They also note that the present day terminology of the WC model is non-arbitratrily based on the current horizontal drawing convention where one strand is placed on top and the other at the bottom.

They also suggested a typical convention which is followed today. If it was due to their suggestion is a different matter altogether, but beyond the scope of this question.


  Given the amount of effort already spent on standardizing such
  databases, and their influence on other disciplines, we feel that the
  genomic definition of Watson and Crick strands has the most mass
  behind it. Specifically, we find that the unambiguous usage of the
  Saccharomyces Genome Database to be the most useful. Under the first
  part of our proposal, the centromere is a reference point that divides
  a chromosome into two arms of unequal length. The chromosome is
  oriented so that shorter arm is on the left and the longer arm on the
  right. Furthermore, the top strand has its 5'-end at the left
  (short-arm) telomere and its 3'-end at the right (long-arm) telomere.
  This strand is the Watson strand. Similarly, the bottom strand has its
  5'-end at the right telomere and its 3' at the left telomere and is
  the Crick strand. We further propose that "top", "forward", and "plus"
  be used as synonyms for the Watson strand and "bottom", "reverse", and
  "minus" for the Crick strand.


and later on...


  If it is ultimately impossible to distinguish Watson and Crick strands
  using biological properties, then we propose that Watson should refer
  to the stand arbitrarily used as a reference in a database (i.e. the
  "plus" stand) and the Crick strand should refer to its complement


But let's come back to sequencing data or FASTQ files. When we align reads, a resulting SAM or BAM file has a column specifying strand information, we usually see a + or - strand. 

Which suggests that the product originated from either the Watson (positive) or the Crick (negative) strands. For example, a read is actually the reverse complement of the product, but since you do a PCR step during you library preparation, that particular bit of information is lost and therefore the downstream analysis protocols tend to consider the entire loci any single product aligns to.

The difference between strand specific sequencing and non-strand specific sequencing are covered here.