Once source shows her surname as REEL, but all others show her as ALEXANDER.
Was one of these surnames her maiden name and the other the name of a prior husband?
Did she also have a prior spouse or partner surnamed BIDDLE?

Descendants of Jacob II are not a Y-DNA match with other descendants of Jacob STROUP I or with at least one of our subject's other sons (viz., Andrew).  They do, however, have a 35/37 match with a BIDDLE.  Normally, I would insist on a higher level match to declare a connection, but their haplotype is rare, reducing the possibility the match could be coincidental.

Given that Jacob II is our subject's eldest child, the most likely explanation is that Jacob II was Catherine's son from a prior marriage (or relationship).  I realize this discovery comes as a tremendous shock to Jacob II's many descendants, but I'm afraid the DNA evidence is undeniable.  Please see this page for further discussion.

— born at Hoyle Creek, Lincoln Co., NC:
  4.  Joseph STROUP, b. 2 May 1776
  5.  John STROUP, b. ca. 1778
  6.  Elizabeth STROUP, b. 1779; d. aft. 1858; m. 15 Feb 1800, Alexander Spencer HEAD
  7.  Andrew STROUP, b. 1782 
  8.  David STROUP, b. 1765-84
  9.  Peter STROUP, b. 1787 
10.  Solomon STROUP, b. 1790

Some sources do not show Nancy as their child.
11.  Nancy STROUP, b. 1795; m. 8 Sep 1811, Jesse ROSS

1.  LDS.  Family Search: Internet Genealogy Service (online):  International Genealogical Index (IGI).

Adam STROUP Birth: 1746, Baltimore, Baltimore, Maryland Father:  Jacob STROUP Source:  patron submission

2.  Brent Holcomb.  1977.  Deed Abstracts of Tryon, Lincoln & Rutherford Counties, North Carolina, 1769-1786, Tryon County Wills & Estates.  Southern Historical Press, Greenville, SC. Number in the first column is the page nubmer in Holcomb; number in the second column is the page number in the deed book.  (In 1779, Tryon Co. was divided into Lincoln and Rutherford Cos.; Tryon Co. ceased to exist.  In 1841, Cleveland Co. was formed from parts of Lincoln and Rutherford Cos.)

p.	p.	Date	Transaction
111	652	16 Nov 1778	ADAM STROUP of Tryon Co., to HENRY TANKERSLEY of Craven Co., S.C., for Þ 70 north currency...land on both sides of Buffalo Creek including his own improvements where he now lives adj. THOMAS WELSHES, 200 A...HENRY CARLOCK (SEAL), Wit: FORNEY NORMAN, GEORGE PATTERSON.  Rec. Apr. term 1784.
100	531- 532	1 June 1779	JACOB STROUP & wf of Lincoln Co., to ADAM STROUP of same, for Þ 74...land on both sides of Sailors branch of Hoyles Creek, 200 A...JACOB STROUP (SEAL), Wit: ANDW NEEL. Rec. Apr. term 1780.
The question in the first transaction is whether "his" refers to Adam or Henry; that is, was Adam living on the land, as owner, or was Henry living on the land, as tenant?  Buffalo Creek is now in Cleveland Co.  For a rough sketch map of the area of Sailor's Branch of Hoyle Creek, most of which is in Gaston Co., click here.

3.  1790 Census Index/Images (online at Ancestry.com — mis-indexed "Shoup"; Image #8 of 11; extracted by Diana Gale Matthiesen):¤•

1790

NC

Lincoln Co.

9th Company

Roll M637_7

p. 123

Ln. L10

Adam Stroup

2-3-4-0

These data indicate:

Gender and Type	Age Class	Therefore Born	Individuals Inferred
2 free white males	16 or over	in or bef. 1774	= Adam (b. 1746) = Jacob (b. 1771) = Michael (b. ca. 1772)
3 free white males	15 or under	1774-1790	= Joseph (b. 1776) = John (b. ca. 1778) = Andrew (b. 1782) = David (b. ca. 1784) = Peter (b. 1787) = Solomon (b. 1788)
4 free white females	all ages	in or bef. 1790	= Catherine (b. 1747) = Katherine (b. 1772) = Elizabeth (b. 1779) = ?

Hard to say why all the numbers are off.  Is this not our subject?  The estimated birthdates of some of the children may be wrong.  Some of the older sons could be married, and there could easily be daughters who died before maturity.  It could also be that the secondary sources, which are all I have for the identity of their children, are way off.

4.  1800 Census Index/Images (online at Ancestry.com; Image #33 of 72 — list was roughly alphabetized; extracted by Diana Gale Matthiesen):¤•

1800

NC

Lincoln Co.

Roll M32_29

p. 857

Ln. 15

Adam Stroup

03001-10101-00

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
3 males	10-15	1784-1790	= Solomon (b. 1788) = Peter (b. 1787) = David (b. ca. 1784)
1 male	45 or over	in or bef. 1755	= Adam (b. 1746)
1 female	9 or under	1790-1800	= Nancy (b. 1795)
1 female	16-25	1774-1784	= Elizabeth (b. 1779)
1 female	45 or over	in or bef. 1755	= Catherine (b. 1747)

5.  1810 Census Index/Images (online at Ancestry.com; Image #2 of 4; extracted by Diana Gale Matthiesen):¤•

1810

NC

Lincoln Co.

Roll M252-40

p. 375A/496

Ln. 13

Adam Stroup

00101-11101-00

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
1 male	16-25	1784-1794	= Solomon? (b. 1788)
1 male	45 or over	in or bef. 1765	= Adam (b. 1746)
1 female	9 or under	1800-1810	= ?
1 female	10-15	1794-1800	= Nancy (b. 1795)
1 female	16-25	1784-1794	= ?
1 female	45 or over	in or bef. 1765	= Catherine (b. 1747)

6.  Census Index:  U.S. Selected Counties, 1820 (Broderbund CD-314):  I checked every "A." or "Adam" in the county and found nothing close.

7.  Census Index:  U.S. Selected Counties, 1830 (Broderbund CD-315):  I checked every "A." or "Adam" in the county and found nothing close.

8.  Revolutionary War Pension Application (photo copy courtesy of Michael STROUPE).  From it we learn that Adam was born in 1746, "within three miles of the City of Baltimore," and that he was living in Lincoln Co., NC, when he enlisted in the North Carolina Line in 1780.

9.  LDS.  Family Search: Internet Genealogy Service (online):  Ancestral File.

10.  Broderbund.  World Family Trees.  Vol. 10, Pedigree No. 1354.

11.  Bill Williamson.  Descendants of Matthias Stroope (online at members.cox.net/wmson/strouppvt.htm).